# Kernel of successive powers of a matrix

For any $n \times n$ matrix $A$, is it true that $\ker(A^{n+1}) = \ker(A^{n+2}) = \ker(A^{n+3}) = \dots$ ? If yes, what is the proof and is there a name to this theorem? If not, for what matrices will it be true? How can I find a counterexample in the latter case?

I know that powers of nilpotent matrices increase their kernel's dimension up to $n$ (for the zero matrix) in the first $n$ steps.

But is it necessary that for all singular matrices, all the rank reduction (if it occurs) must be in the initial exponents itself? In other words, is it possible for some matrices to have $\ker(A^{k}) = \ker(A^{k+1}) < \ker(A^{k+1+m})$ for some $m,k > 0$?

• The title isn't supposed to replace the first line of your question. As for the question, the answer depends on how you quantify over $n$. – Git Gud Sep 10 '14 at 18:59
• Added the first line. Could you please explain what you mean by 'quantify over n'? – allrtaken Sep 10 '14 at 19:07
• Let $P(n)$ be expression in the title before. If you mean $\exists n\in \mathbb NP(n)$, then the statement is true. If you mean $\forall n\in \mathbb NP(n)$, then the statement is false. – Git Gud Sep 10 '14 at 19:12
• I meant n to be the dimension of the matrix. – allrtaken Sep 10 '14 at 19:21

This is true. To my knowledge, there is no name for this theorem.

You can think of this as a consequence of Jordan canonical form. In particular, we can always write $$A = S[N \oplus P]S^{-1}$$ Where $N$ is nilpotent and $P$ has full rank. It suffices to show that $N$ has order of nilpotence at most equal to $n$, and that $P$ never reduces in rank.

• I want to check if I understood this correctly. Does the matrix N comprise of Jordan blocks with eigenvalue 0, and P is the matrix comprising of Jordan blocks corresponding to the other eigenvalues? – allrtaken Sep 10 '14 at 19:21
• @allrtaken that's exactly right. It is useful to note that a matrix is nilpotent if and only if all of its (complex) eigenvalues are equal to $0$. – Omnomnomnom Sep 10 '14 at 20:25
• I am not sure whether this is the right place for asking this question, but can you suggest a good textbook that covers the theory around this? – allrtaken Sep 11 '14 at 18:26
• Most linear algebra texts geared towards advanced undergraduates or graduates cover Jordan Canonical form at some point. I, in particular, used Horn and Johnson. Axler's "Linear Algebra Done Right" might be another good bet. If you want some more ideas, you could always post another question on this site. – Omnomnomnom Sep 11 '14 at 19:01

An important observation to be made here is that the if for some $k$, we have $\ker(A^k) = \ker(A^{k+1})$, then $\forall j\geq 0, \ker(A^{k+j}) = \ker(A^k)$. To show this, it would be sufficient to show that $\ker(A^{k+2}) = \ker(A^{k+1})$, and the rest would follow from a simple inductive argument.

Note that, we have $\ker(A^{k+1}) \subseteq \ker(A^{k+2})$, and thus it is enough to show that $\ker(A^{k+1}) \supseteq \ker(A^{k+2})$.

For this, consider a vector $v$ such that $v \in \ker(A^{k+2})$, i.e., $A^{k+2}v = 0$. Then, $Av \in \ker(A^{k+1})$ because $A^{k+1}(Av) = 0$. Since $\ker(A^{k+1}) = \ker(A^k)$, we have $Av \in \ker(A^k)$. Thus, $A^{k}(Av) = 0$, and hence $A^{k+1}v = 0$, which implies that $v \in \ker(A^{k+1})$.

Clearly, $\ker(A^{k+2}) \subseteq \ker(A^{k+1})$, and thus $\ker(A^{k+2}) = \ker(A^{k+1})$


Relative to the air, the paper is moving downwards, and so there will be an upward resistive force on the paper. https://study.com/academy/lesson/air-resistance-and-free-fall.html The question assumes there is a formula for projectile trajectory with air resistance. Q.1: A plane moving with a velocity of $$50 ms^{-1}$$ , … object is opposed by the aerodynamic If gravity is the only influence acting upon various objects and there is no air resistance, the acceleration is the same for all objects and is equal to the gravitational acceleration 9.8 meters per square second (m/s²) or 32.2 feet per square second (ft/s²) … 4) density of the falling object is considerably high. There is a large resultant force and the object accelerated quickly. For the ideal situations of these first few chapters, an object falling without air resistance or friction is defined to be in free-fall. The mass of an object contributes to two different phenomena: Gravity and inertia. Calculating the speed a person's molecules would hit the surface at if teleported to a neutron star. Suppose, further, that, in addition to the force of gravity, the projectile is subject to an air resistance force which acts in the opposite direction to its … The equations ignore air resistance, which has a dramatic effect on objects falling an appreciable distance in air, causing them to quickly approach a terminal velocity. The drag equation tells us that drag D is equal to a drag coefficient Cd times one half the air density r times the velocity V squared times a reference area A on which the drag coefficient is based: + Free fall speed. For objects which move slowly relative to the air (such as falling dust particles), the resistive force is directly proportional to the object’s velocity  relative to air. surface of the earth. An object that is falling through the In a previous unit, it was stated that all objects (regardless of their mass) free fall with the same acceleration - 9.8 m/s/s. Roughly: $F_D = \frac{C_D\rho A v^2}{2}$ where F D is the drag force, C D is the drag coefficient, ρ is the density of air, A is the cross-sectional area of the ball (a regulation baseball has a circumference between 9 and 9.25 inches), and v is the velocity of the ball. I am applying air resistance to a falling sphere using the Euler method. Using the impact force calculator. This formula is having wide applications in aeronautics. 3 Calculate the downward pull of gravity. Calculate impulse need on object to throw i Y meters into the air, with varying mass. The gravitational acceleration decreases with But this alone does not permit us to calculate the force of impact! Viewed 26k times 2. The default value of the air resistance coefficient, k=0.24(kg/m), assumes the value in skydiving. Yes! If the value of the constant  4.0×10-11 kg s-1, find the terminal velocity. When an object is dropped from a height and that in vacuum then this free fall is observed actually! The other force is the air resistance, or drag of the object. Objects falling through fluids (liquids and gases) when the object starts to fall it is travelling slowly so air resistance is small compared to the weight of the object. 5) shape of the object is such (aerodynamic) that it cuts through air without much resistance. Close enough to the earth to encounter air resistance, this acceleration is 9.8 meters per second squared, or 32 feet per second squared. + Equal Employment Opportunity Data Posted Pursuant to the No Fear Act 1 $\begingroup$ This … 0. I was wondering where I might look to get some simplified math to calculate the amounf of air resistance on a falling object if I know the shape, mass, and volume of the object. The acceleration of free-falling objects is therefore called the acceleration due to gravity. The acceleration due to gravity is constant, which means we can apply the kinematics equations to any falling object … For an object that falls for 0.850 seconds, the v = 9.81 m/s^2 * 0.850 s = 8.34 m/s. Our team is working on the payload for a student rocket competition. A falling object will reach a constant speed when there is a restraining force, such as drag from the air. Example: A stone is to be dropped from … The expressions will be developed for the two forms of air drag which will be used for trajectories: although the first steps will be done with just the form -cv 2 for simplicity. The force with which the falling object is being pulled down equals the object's mass times acceleration due to … of the object, and the second force is the aerodynamic The kinetic energy just before impact is equal to its gravitational potential energy at the height from which it was dropped: K.E. Eventually, the body reaches a speed where the body’s weight is exactly balanced by the air resistance. F = force due to air resistance, or drag (N) k = a constant that collects the effects of density, drag, and area (kg/m) v = the velocity of the moving object (m/s) ρ = the density of the air the object moves through (kg/m 3) C D = the drag coefficient, includes hard-to-measure effects (unitless) A = the area of the object the air presses … Without the effects of air resistance, the speed of a body that is free-falling towards the Earth would increase by approximately 9.8 m/s every second. ... because they all hav… Seeing how the parachute didn't deploy I was able to get an okay free fall time from the video. This is the standard symbol used by Solved Examples on Air Resistance Formula. Note: In reality, the calculation is not so simple, with many other factors also coming into play. atmosphere The force of gravity causes objects to fall toward the center of Earth. = J. 2. difference For the ideal situations of these first few chapters, an object falling without air resistance or friction is defined to be in free-fall. Formula to calculate terminal velocity. Evaluate air resistance constant using Monte Carlo method. Terminal velocity is constant and its unit is meter per second. Air Resistance: the physics of how objects fall with air resistance. vector quantities. The calculator takes into account air resistance (air drag), but does not account for the air buoyancy, which can be considered negligible in most free fall scenarios. The drag coefficient is a function of things like surface roughness, ball speed, and spin, varying between 0.2 and 0.5 for speeds … Newton's In case of larger objects at higher velocities, the force of air resistance (F air) is given as, Fair = -½cρAv2 is subjected to two external Our problem, of course, is that a falling body under the influence of gravity and air resistance does not fall at constant speed; just note that the speed graph above is not a horizontal line. The default value of the air resistance coefficient, k=0.24(kg/m), assumes the value in skydiving. [6] 2019/11/27 23:10 Male / 30 years old level / An office worker / A public employee / Useful / Purpose of use Falls off the side of a freeway for four seconds before hitting the ground [7] 2019/08/13 01:54 Male / 20 years old level / A teacher / A researcher / Useful / Purpose of use Calculate the depth of the steel … But in the atmosphere, the motion of a falling object is opposed by the air resistance, or drag. velocity V and Accessibility Certification, + Equal Employment Opportunity Data Posted Pursuant to the No Fear Act, + Budgets, Strategic Plans and Accountability Reports. The effect of air resistance varies enormously depending on the size and geometry of the falling object—for example, the equations are hopelessly wrong for a feather, which has a low mass but offers a large resistance to the air. So, we can write: The value of  depends on the shape and size of the body. where the drag is exactly equal to the weight. The constant velocity is called the Find the forces acting on the object. This resistive force is called air resistance. Air resistance increases with surface area, but also with velocity, because a higher velocity means an object is displacing a greater volume of air per second. This formula is having wide applications in aeronautics. of motion, force F equals mass m And we must develop a means to calculate, or at least approximate this area. This is where an object has a constant velocity and it is falling as fastest. of motion. In keeping with the scientific order of operations, you must calculate the exponent, or t^2 term, first. Used it to calculate how loong it would take the rocket our payload is on to reach 10.000 meters. Terminal velocity is constant and its unit is meter per second. Furthermore, the distance traveled by a falling object (d) is calculated via d = 0.5gt^2. Solved Examples on Air Resistance Formula. Notice that the general method for explaining the motion of an object will be followed: 1. Q: How do you calculate air resistance? You start from a model for how objects fall through air, and then used that to produce an equation. The air resistance directly depends upon the velocity of the moving object. Physics Ninja looks at a problem of air resistance during free fall. For the example from Step 1, t^2 = 2.35^2 = 5.52 s^2. The terminal velocity for objects moving fast in air can be given by . At terminal velocity, the downward force is equal to the upward force, so mg = –bv or mg = –cv2, depending on whether the drag force follows the first or second relationship. The acceleration is constant when the object is close to Earth. As the object accelerates the air resistance increases but the weight stays the same, so the resultant force is not as great as before but the object is still accelerating. Free fall with air resistance (distance and velocity) Calculator [10] 2020/10/01 08:36 Male / 40 years old level / An engineer / Useful / Purpose of use Made a homemade rocket with my kids (no store parts purchased, thanks YouTube!). this would be the only 0. A freely falling object will be presumed to experience an air resistance force proportional to the square of its speed. But in the atmosphere, the motion of a falling object is opposed by the air resistance, or drag. We assume that there are two forces affecting the vertical descent of the object: gravity and air resistance. It quickly reaches a point A. by interpretive software. As a consequence, gravity will accelerate a falling object so its velocity increases 9.81 m/s or 32 ft/s for every second it experiences free fall. Air resistance causes objects to fall at hit terminal velocity. In fact, we even have a value for this acceleration: g , or 9.8 m / s ^2. Without the effect of air resistance, each object in free fall would keep accelerating by 9.80665 m/s (approximately equal to … For a complete index of these free videos visit http://www.apphysicslectures.com the square of the distance from the center of the earth. If the mass of an object remains constant, the motion of the object can be described by Newton's second law of motion, force F equals mass m times acceleration a : F = m * a Difference Between Acceleration and Deceleration, Difference Between Sonogram and Ultrasound, What is the Difference Between Dependency Theory and Modernization Theory, What is the Difference Between Oak and Birch, What is the Difference Between Model and Paradigm, What is the Difference Between Cassoulet and Casserole, What is the Difference Between Palm Sugar and Cane Sugar, What is the Difference Between Nation and Nation State. Now, Newton's laws point out that light and heavy objects will fall with the same velocity. 1. When you’re calculating force for a falling object, there are a few extra factors to consider, including how high the object is falling from and how quickly it comes to a stop. on which the drag coefficient is based: On the figure at the top, the density is expressed by the Greek symbol Calculate the terminal velocity of a human body (e.g. "rho". The air constant, or the drag coefficient of the object, is dependent on the shape of the object and is a dimensionless quantity. I know how to calculate the distance of a free falling object without air resistance, d = 1/2g*t² and I do not know how I would include air resistance in there $\endgroup$ – Marvin Johanning Sep 16 '16 at 15:55 forces. of the object times the gravitational acceleration For the general case of a projectile fired with velocity v, at an angle α to the horizontal there is no analytic solution. Used this calculator to determine how high the rocket went. defines the weight W to be If the object deforms when it makes impact – a piece of fruit that smashes as it hits the ground, for example – the length of the portion of the object that deforms can be used as … The hardest part to work out when you calculate falling object forces is the distance traveled. How to Calculate Air Resistance of a Falling Object when the Object Falls Slowly in Air For objects which move slowly relative to the air (such as falling dust particles), the resistive force is directly proportional to the object’s velocity relative to air. Air Resistance: the physics of how objects fall with air resistance. One of the reasons this problem is so challenging is that, in general, there are many different forces acting on such objects, including: gravity; drag; lift ; thrust; … + The President's Management Agenda See, for example, Figure 6. Ron Kurtus' Credentials. It can also be determined by the change in potential energy of the object due to gravity. The net external force is then equal to the From the definition of velocity, we can find the velocity of a falling object is: g is the free fall acceleration (expressed in m/s² or ft/s²). The motion of an object though a fluid is one of the most complex problems in all of science, and it is still not completely understood to this day. But for most practical problems in the atmosphere, we can assume this Charged Particle Motion in Up: Multi-Dimensional Motion Previous: Motion in a Two-Dimensional Projectile Motion with Air Resistance Suppose that a projectile of mass is launched, at , from ground level (in a flat plain), making an angle to the horizontal. times the The speed and the altitude of a free-falling object are defined as follows: where. These models can get arbitrarily complicated, depending on how much you want to just model from scratch and how much you are prepared to measure. 7. Air resistance also known as drag force, is the force which opposes the relative motion of the objects in the air. For example, it can be used to calculate the impact force of a vehicle (car, truck, train), plane, football, of birds hitting a plane or wind mill, as well as for falling bodies that crash into the ground. This is true because acceleration is equal to force divided by mass. A golf ball falling in air has a drag coefficient of 0.26. of the weight and the drag forces: The acceleration of the object then becomes: The drag force depends on the square of the velocity. Were it not for air resistance, all free-falling objects would fall at the same rate of acceleration, regardless of their mass. times acceleration a: We can do a little algebra and solve for the And the gravitational force is only slightly larger than the air resistance force. Terminal velocity is the steady speed achieved by an object freely falling through a gas or liquid. Determine the net force acting on the object and B. calculate the acceleration of the object. When falling, there are two forces acting on an object: the weight, mg, and air resistance, –bv or –cv2. The heavier an object is, the stronger its resistance to an accelerating force will be: Heavier objects are harder to set in motion, … Determine how terminal velocity of a falling object is affected by air resistance and mass (from Physics with Vernier, experiment 13). Here, the body has reached terminal velocity, . Work by … Here is a common way to calculate the magnitude of the drag force on a moving object. The acceleration of free-falling objects is therefore called the acceleration due to gravity. Terminal velocity occurs when the resistance of the air has become equal to the force of gravity. A basic problem where a falling object is subject to an air resistance bv. The work done equals the product of the force of gravity and the displacement of the object. So as the body accelerates its velocity and the drag increase. However, for the sake of this example, we have assumed that the only factors which affect the pollen grain’s fall are gravity and air resistance, and the air resistance is also assumed to be directly proportional to the grain’s velocity. + Budgets, Strategic Plans and Accountability Reports For a complete index of these free videos visit http://www.apphysicslectures.com terminal velocity. + Non-Flash Version The acceleration due to gravity is constant, which means we can apply the kinematics equations to any … You can estimate this to come up with an answer, but there are some situations where you can put together a firmer figure. If allowed to free fall for long enough, a falling object will reach a speed where the force of the drag will become equal the force of gravity, and the … Calculate the distance the object fell according to d = 0.5 * g * t^2. Air Resistance Formula is helpful in finding the air resistance, air constant, and velocity of the body if the remaining numeric are known. acceleration of the object in terms of the net external Given that it has an effective cross-sectional area of 1.4×10-3 m2, find the air resistance on the ball when the ball is moving at a speed of 20 m s-1. The paper does not … Terminal velocity occurs when the resistance of the air has become equal to the force of gravity. g: the value of g is 9.8 meters per square second on the Projectiles with air resistance. The force applied by gravity near to a massive body is mostly constant, but forces like air resistance increase the faster the falling object goes. For many things, air resistance faced while falling down creates a force that pushes the it upwards and slows the descent speed. If an object of mass m= kg is dropped from height h = m, then the velocity just before impact is v = m/s. Active 8 years, 1 month ago. But in the atmosphere, the motion of a falling Contact Glenn. First, the effects of air drag are often small when dealing with falling balls and rolling carts (a staple of intro physics labs). The heavier an object is, the stronger the gravitational pull it experiences. This calculator calculates how fast you're moving after falling a certain distance — your free fall speed.It ignores friction (air, rock, rope, or otherwise) and relativistic effects: We hope that you won't fall far enough to have either of these make much of a difference! When drag is equal to weight, there is no net external force In either case, since g and b or c are constants, the terminal velocity is affected by the mass of the object. Air resistance is also called "drag", and the unit for this force is Newtons (N). How to Calculate Air Resistance of a Falling Object, Difference Between Hardness and Toughness, Difference Between Attenuation and Absorption. Websites. Whenever objects move relative to the air, the objects experience a resistive force which is in the opposite direction to the body’s velocity relative to the air. Air has a much greater effect on the motion of the paper than it does on the motion of the baseball. drag equation Consider a spherical object, such as a baseball, moving through the air. drag. In the text below, we will explain how this tool works. Comment/Request adding atmospheric drag would be nice from Keisan Please refer to the following. aeronautical engineers. drag coefficient Cd Air resistance can be calculated by multiplying air density by the drag coefficient, multiplied by area all over two, and then multiplied by velocity squared. weight equation So, we can write: The value of depends on the shape and size of the body. Velocity of a Falling Object: v = g*t. A falling object is acted on by the force of gravity: -9.81 m/s 2 (32 ft/s). The baseball is still accelerating when it hits the floor. squared times a reference g is the acceleration due to gravity ( 9.8N/Kg ). Calculate the metric solution of velocity by multiplying the time in free fall by 9.81 m/s^2. 3) area of the surface facing air resistance is small. second law Calculates the free fall distance and velocity with air resistance from the free fall time. Text Only Site area A The object then falls at a constant velocity as described by Objects falling on the ground at low speed can be considered being in free fall because in this case the air resistance is negligible and can be neglected. The The symbol looks like a script "p". Free falling of object with no air resistance [duplicate] Ask Question Asked 9 years, 10 months ago. Calculates the free fall energy and velocity without air resistance from the free fall distance. acting on the object. Compare the falling of a baseball and a sheet of paper when dropped from the same height. 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how to calculate air resistance of a falling object 2021


# How does the middle term of a quadratic $ax^2 + bx + c$ influence the graph of $y = x^2$?

Every parabola represented by the equation $$y = ax^2 + bx + c$$ can be obtained by stretching and translating the graph of $$y = x^2$$.

Therefore:

The sign of the leading coefficient, $$-a$$ or $$a$$, determines if the parabola opens up or down i.e.

The leading coefficient, $$a$$, also determines the amount of vertical stretch or compression of $$y = x^2$$ i.e.

The constant term, $$c$$, determines the vertical translation of $$y = x^2$$ i.e.

Now for $$bx$$. Initially, I thought it would determine the amount of horizontal translation since the constant term, $$c$$, already accounted for the vertical translation, but when I plugged in some quadratics the graph of $$y = x^2$$ translated both horizontally and vertically. Here are the graphs:

Seeing as the middle term, $$bx$$, does more than just horizontally translate, how do you describe its effect on $$y=x^2$$? Would it be accurate to say that it both horizontally and vertically translates the graph of $$y = x^2$$?

• +1 for beautiful graphs and your efforts too!! – StammeringMathematician Sep 27 '18 at 4:18
• @StammeringMathematician Thank you! I used this to make the graphs: desmos.com/calculator – Slecker Sep 27 '18 at 4:22
• +1 from me as well. This attitude should be highly encouraged here on MSE. – Ahmad Bazzi Sep 27 '18 at 4:34
• Slecker. Beautiful +. – Peter Szilas Sep 27 '18 at 9:17

Yes, it will effect both a horizontal and vertical translation, and you can see how much by completing the square. For example, $$x^2+3x=\left(x+\frac32\right)^2-\frac94$$

Compare that to your graph of $$y=x^2+3x$$. Of course, if the coefficient of the quadratic term is not $$1$$ things get a little more complicated, but you can always see what the graph the graph will look like by completing the square.

• It took me a while to realize that you transformed it into vertex form. So would the reason that $bx$ affects both a horizontal and vertical translation be because it occurs in both the x and y-coordinates of the vertex, since the vertex coordinates are ($\frac{-b}{2a}$, $\frac{ -b^2+4ac}{4a})$? – Slecker Sep 27 '18 at 4:46
• @Slecker I'm not familiar with the term "vertex form," but I would say that you are correct. – saulspatz Sep 27 '18 at 5:04

Look at $$2$$ Cartesian coordinate systems $$X,Y$$ and $$X',Y'$$.

Origin of $$X',Y$$' is located at $$(x_0,y_0)$$, $$X'$$-axis parallel $$X$$-axis , $$Y'$$-axis parallel $$Y$$-axis(a translation),i.e.

$$x= x_0+x'$$; $$y= y_0+ y'$$.

Set up your normal parabola in the $$X',Y'$$ coordinate system.

$$y'=ax'^2$$, vertex at $$(0',0')$$.

Revert to original $$x,y$$ coordinates .

$$y-y_0= a(x-x_0)^2$$ ;

$$y=ax^2 -2(ax_0)x +ax_0^2$$.

Compare with $$y =ax^2+bc +c$$:

$$b=-2ax_0$$.

Can you interpret?

• I'm having a hard time understanding what you mean by "Revert to original $x$, $y$ coordinates" and where the subsequent equation, $y-y_0 = a(x-x_0)^2$, came from. I think once I understand that I can interpret the rest of your answer. – Slecker Sep 27 '18 at 15:22
• Slecker.Draw two coordinate systems, x,y and another one ,call it x',y'.Say, you put the origin of the x',y' system at x_0=3, y_0=4.x'y' system has its origin at (x_0,y_0)=(3,4), ok?. put a normal parabola y'=ax'^2 in the x',y' system.x'=1; y'=a; everything in x'y'.Take any x' coordinate, say x'=7, what is the x value in the original system: x= 7+ 3= x' +x_0 ok? Likewise y= y'+y_0. Solve for x' and y' and plug into y'=ax'2, get (y-y_0)=a(x-x_0)^2, now you are back in the original system.Your b =-2ax_0, where x_0 is the x-coordinate of the vertex.Let me know if ok. – Peter Szilas Sep 27 '18 at 17:56
• Ah ok thanks for the clarification! – Slecker Sep 27 '18 at 18:26
• Slecker. If anything else, just say so:) – Peter Szilas Sep 27 '18 at 18:35


# Cone shaped related rates of change question

A container is in the shape of a cone of semi-vertical angle $30^\circ$, with it's vertex downwards.

Liquid flows into the container at ${{\sqrt {3\pi } } \over 4}{\rm{ }}c{m^{^3}}/s$

At the instant when the radius of the circular surface of the liquid is 5 cm, find the rate of increase of:

(a) The radius of the circular surface of the liquid

(b) The area of the circular surface of the liquid

My attempt:

${{dV} \over {dt}} = {{\sqrt {3\pi } } \over 4}$

(A) I need to find the rate at which the radius increases as "h" increases, so I have to find ${{dr} \over {dt}}$.

The equation for the volume of a cone is:

$V = {1 \over 3}\pi {r^2}h$

I now must form a function in terms of r for h. As we are asked to a compute when the radius is 5 we can form an equation using similar triangles, so:

\eqalign{ & {h \over x} = {r \over 5} \cr & x: \cr & \tan 30^\circ = {5 \over x} \cr & x = {5 \over {\tan 30^\circ }} \cr & x = 5\sqrt 3 \cr & so: \cr & h = r\sqrt 3 \cr}

\eqalign{ & V = {1 \over 3}\pi {r^2}(r\sqrt 3 ) \cr & V = \pi {r^3}{{\sqrt 3 } \over 3} \cr & {{dV} \over {dr}} = \pi {r^2}\sqrt 3 \cr & {{dr} \over {dt}} = {{dV} \over {dt}} \times {{dr} \over {dV}} \cr & {{dr} \over {dt}} = {{\sqrt {3\pi } } \over 4} \times {1 \over {\pi {r^2}\sqrt 3 }} \cr & {{dr} \over {dt}} = {{\sqrt {3\pi } } \over {\pi {r^2}4\sqrt 3 }} = {{\sqrt \pi } \over {\pi {r^2}4}} \cr}

\eqalign{ & r = 5: \cr & {{\sqrt \pi } \over {4(25)\pi }} = {{\sqrt \pi } \over {100\pi }} = 0.005641... \cr}

For part (A) the answer is stated as 0.01 cm/s, nowhere in the question have I been asked to round my answer, have I obtained the correct answer? I just want to make sure..

Part (b)

Part (B) requires that I calculate the rate of increase of the circular area of the liquid, so essentially ${{dA} \over {dt}}$.

Area of a circle is: $A = \pi {r^2}$

\eqalign{ & A = \pi {r^2} \cr & {{dA} \over {dr}} = 2\pi r \cr & {{dA} \over {dt}} = {{dr} \over {dt}} \times {{dA} \over {dr}} \cr & {{dA} \over {dt}} = {{\sqrt \pi } \over {4\pi {r^2}}} \times 2\pi r \cr & {{dA} \over {dt}} = {{\sqrt \pi } \over {2r}} \cr & r = 5: \cr & {{dA} \over {dt}} = 0.1\sqrt \pi {\rm{ c}}{{\rm{m}}^2}/s \cr}

However the answer in the book for b is ${{dA} \over {dt}} = 0.1\pi {\rm{ c}}{{\rm{m}}^2}/s$ (pi is not square rooted), Where have I gone wrong?

Furthermore I'd love it if any answerers could suggest how I could improve on how I've done things, and any tricks/tips that would make things easier for myself in the future.

Thank you.

• If you are going to use $x$, you should define it. In fact, you don't need to find $h$, but if you want to, $\frac rh=\tan 30^\circ=\frac {\sqrt 3} 3$, so $h=5\sqrt 3$, which you got. – Ross Millikan May 2 '13 at 22:37
• @RossMillikan, Oh okay, I'll bear that in mind, thank you! – seeker May 2 '13 at 22:54

"Liquid flows into the container at: $\;\displaystyle \dfrac{\sqrt{3}\,\pi}{4} = {{(\sqrt {3})\,\pi} \over 4}{\rm{ }}\text{ cm$^3$per sec}$"
This would explain both discrepancies, (between both the solutions you obtained, and the solutions of the text), since your work seems to be fine, and was clearly done carefully. Simply changing your final evaluations using $\pi$ instead of $\sqrt{\pi}$ will yield the solutions you are given.


1. watchmath

The term of the absolute series is $$|\arctan(1/n)|/n^2<\frac{\pi/2}{n^2}$$. But the series $$\sum 1/n^2$$ is convergent (since it is a p-series with p=2), so the absolute series converges. Then the series is absolutely convergent.

2. anonymous

how did you come up with pi/2/x^2

3. watchmath

the range of the arctan function is between -pi/2 and pi/2

4. anonymous

and why did you put over x^2

5. watchmath

well I just you that arctan (1/n) < pi/2. The n^2 is already on the denominator on the first place.

6. anonymous

so would you always compare it to pi/2 divided by denominator given in the problem

7. anonymous

for tan

8. watchmath

I wouldn't say always. I just see the opportunity that if we compare with pi/2 then I can have some conclusion. It is possible for a problem to have an arctan bu we don't compare with the pi/2.

9. anonymous

and if there isnt anything in the denominator of the given problem you would compare it to pi/2

10. watchmath

No, I won't if there is nothing on the bottom, I can compare arctan(1/n) < pi/2. But the series $$\sum pi/2$$ is divergent. So comparing with pi/2 doesn't give me any conclusion.

11. anonymous

okay what would you comoare arcsin (1/n) to and 1-cos1/n to

12. watchmath

the term of the series only arcsin(1/n) ?

13. anonymous

yah

14. watchmath

Just to make sure. So your series is $$\sum_{n=1}^\infty \arcsin(1/n)$$? I can't think how to do this right away...

15. anonymous

yah

16. watchmath

ok, arcsin is an increasing function and for positive x we have x > sin x Apply the arcsin arcsin x > x It follows that arcsin(1/n) > 1/n But the harmonic series $$\sum 1/n$$ is divergent Hence $$\sum \arcsin(1/n)$$ is divergent as well.

17. anonymous

what if it was just sin

18. watchmath

sin what?

19. anonymous

if instead of arc sin it was sin would you still make the same comparison

20. watchmath

you mean sin(1/n) ?

21. anonymous

yah

22. watchmath

well we can't compare to 1/n since 1/n > sin(1/n) So I don't know the answer yet.

23. anonymous

oh okay

24. anonymous

also if you were given sigma 1-cos(1/n) what would you compare it to

25. watchmath

That is actually a nice problem. I'll post it as a new question so everybody can give a response (BTW it is convergent)


# Power set of a set with an empty set

When a set has an empty set as an element, e.g.$\{\emptyset, a, b \}$. What is the powerset?

Is it: $$\{ \emptyset, \{ \emptyset \}, \{a\}, \{b\}, \{\emptyset, a\} \{\emptyset, b\}, \{a, b\}, \{\emptyset, a, b\}\}$$

Or

$$\{ \emptyset, \{a\}, \{b\}, \{\emptyset, a\} \{\emptyset, b\}, \{a, b\}, \{\emptyset, a, b\}\}$$

Or $$\{ \{\emptyset\}, \{a\}, \{b\}, \{\emptyset, a\} \{\emptyset, b\}, \{a, b\}, \{\emptyset, a, b\}\}$$

The confusion arises for me because, the powerset of every non-empty set has an empty set. Well the original set already has the empty set. So we don't need a subset with an empty set.

Somehow, the first one seems correct. Yet, I can't seem to accept it.

• The first one: $\;\emptyset\;$ is one of the elements of the given set, besides being a subset of it. – DonAntonio Oct 12 '16 at 12:39
• Let $c$ denote $\varnothing$. What is the power set of $\{a,b,c\}$? Now write $\varnothing$ instead of $c$ again. – Asaf Karagila Oct 12 '16 at 13:16

The first one is correct.

This is because $\emptyset$ and $\{\emptyset\}$ are different. The first is an empty set whereas the second is a set whose only element is the empty set.

Both are subsets of the given set. This is because the $\emptyset$ is the subset of every set, and as it happens to be an element of the given set, the set containing it as its element is also its subset.

If a set $A$ is such that $\emptyset\in A$, its power set must necessarily contain these two sets:

• $\emptyset$ (like all other power sets), corresponding to selecting nothing from $A$ (not even $\emptyset$, which is something)
• $\{\emptyset\}$, corresponding to selecting $\emptyset$ only

Therefore only the first of your proposed answers is correct, as you think.

Your suggestions differ by having $\emptyset$ and/or $\{\emptyset\}$ included or not.

• We have $\emptyset\in\mathcal P(X)$ because $\emptyset\subseteq X$ (which would hold for any other $X$ as well)
• We have $\{\emptyset\}\in\mathcal P(X)$ because $\{\emptyset\}\subseteq X$ (which is the case because $\emptyset\in X$ in this specific problem)

Therefore, your first variant is correct (and the other two are incorrect because $\emptyset\ne\{\emptyset\}$).

• Your second bullet is strangely phrased to me; we have $\{\emptyset\}\in\mathcal{P}(X)$ simply because $\emptyset\in X$ in this specific problem. The fact that $\emptyset\subseteq X$, which is true for every set $X$, has nothing to do with it? – Inactive - avoiding CoC Oct 12 '16 at 15:47


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# What is the average (arithmetic mean) of eleven consecutive

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What is the average (arithmetic mean) of eleven consecutive integers?

(1) The average of the first nine integers is 7
(2) The average of the last nine integers is 9
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What is the average (arithmetic mean) of eleven consecutive integers?

Consecutive integers represent evenly spaced set. For every evenly spaced set mean=median, in our case $$mean=median=x_6$$.

(1) The average of the first nine integers is 7 --> $$x_1+x_2+...+x_9=63$$ --> there can be only one set of 9 consecutive integers to total 63. Sufficient.

If you want to calculate: $$(x_6-5)+(x_6-4)+(x_6-3)+(x_6-2)+(x_6-1)+x_6+(x_6+1)+(x_6+2)+(x_6+3)=63$$ --> $$x_6=8$$.

OR: Mean(=median of first 9 terms=5th term)*# of terms=63 --> $$x_5*9=63$$ --> $$x_5=7$$ --> $$x_6=7+1=8$$

(2) The average of the last nine integers is 9 --> $$x_3+x_4+...+x_{11}=81$$ --> there can be only one set of 9 consecutive integers to total 81. Sufficient.

If you want to calculate: $$(x_6-3)+(x_6-2)+(x_6-1)+x_6+(x_6+1)+(x_6+2)+(x_6+3)+(x_6+4)+(x_6+5)=81$$ --> $$x_6=8$$.

OR: Mean(=median of last 9 terms=7th term)*# of terms=81 --> $$x_7*9=81$$ --> $$x_7=9$$ --> $$x_6=9-1=8$$

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Re: If 11 consecutive integers are listed from least to [#permalink]

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08 Mar 2013, 05:51
so its not possible to have a list of numbers with positive and negative numbers?
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Re: If 11 consecutive integers are listed from least to [#permalink]

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08 Mar 2013, 06:02
fozzzy wrote:
What is the average (arithmetic mean) of eleven consecutive integers?

(1) The average of the first nine integers is 7.
(2) The average of the last nine integers is 9.

so its not possible to have a list of numbers with positive and negative numbers?

How it is possible? From both statements it follows that the set is {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}.
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Re: What is the average (arithmetic mean ) of eleven consecutive [#permalink]

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zz0vlb wrote:
What is the average (arithmetic mean ) of eleven consecutive integers?

(1) The avg of first nine integers is 7
(2) The avg of the last nine integers is 9

Here is a neat little trick for such kind of problems:

Will be better illustrated using a numerical example: take the set {2,3,4,5,6}. Here the common difference (d)=1. The initial average = 4. Now, the averge of the set, after removing the last integer of the set(i.e. 6)will be reduced by exactly $$\frac{d}{2} units \to$$ The new Average = $$4-\frac{1}{2} = 3.5$$

Again, for the new set of {2,3,4,5} the average is 3.5 . Now, if the last integer is removed, the new average will again be = 3.5-0.5 = 3.

Similarly, for the same set {2,3,4,5,6}, if we remove the first integer from the given set, the average increases by 0.5 and so on and so forth.

Back to the problem:

From F.S 1, we know that the average of the first 9 integers is 7. Thus, the average with the original 11 integers must have been 7+0.5+0.5 = 8. Sufficient.

From F.S 2, we know that the average of the last 9 integers is 9, thus the average with the initial 11 integers must have been 9-0.5-0.5 = 8. Sufficient.

D.
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Re: What is the average (arithmetic mean ) of eleven consecutive [#permalink]

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zz0vlb wrote:
What is the average (arithmetic mean ) of eleven consecutive integers?

(1) The avg of first nine integers is 7
(2) The avg of the last nine integers is 9

As a general rule whenever there is a AP the average of the series is always the median of the series. Here it is a AP with difference 1

1. First 9 integers average is 7 . So the median that is the 5th digit is 7. Hence we can easily find the series and the average of the 11 consecutive digit series. Sufficient
2. Average of last 9 integers is 9 hence we know that for this subset of 9 integers the 5th integer would be 9 and we can find the series on the basis of this and the average. Sufficient

And is D
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Re: What is the average (arithmetic mean) of eleven consecutive [#permalink]

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09 Mar 2015, 19:15
I considered following approach

if the smallest number in set is x , then sum of 11 consecutive numbers = 11x+(1+2+...10)=11x+55--->A
if largest number in set is x ,then sum of 11 consecutive numbers=11x-(1+2+10)=11x-55

Now as per statement 1 , average of first 9 numbers is 7 i.e sum =63
sum of 11 numbers =63+x+9+x+10----->B

Equating A& B
11X+55=63+X+9+10 ,which can be solved to get x=3

statement I is sufficient
similar approach for Statement II
11X-55=8+2X-19 ,can be solved to get X=13

statement 2 is sufficient

OA=D
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Re: What is the average (arithmetic mean) of eleven consecutive [#permalink]

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09 Mar 2015, 19:26
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Hi All,

When you look at this question, if you find yourself unsure of where to "start", it might help to break down everything that you know into small pieces:

1st: We're told that we have 11 consecutive integers. That means the 11 numbers are whole numbers that are in a row. If we can figure out ANY of the numbers AND it's place "in line", then we can figure out ALL of the other numbers and answer the question that's asked (the average of all 11 = ?)

2nd: Fact 1 tells us that the average of the FIRST 9 integers is 7. For just a moment, ignore the fact that there are 9 consecutive integers and let's just focus on the average = 7.

What would have to happen for a group of consecutive integers to have an average of 7?

Here are some examples:

7

6, 7, 8

5, 6, 7, 8, 9

Notice how there are the SAME number of terms below 7 as above 7. THAT'S a pattern.

With 9 total terms, that means there has to be 4 above and 4 below:

3, 4, 5, 6,.......7.......8, 9, 10, 11

Now we have enough information to figure out the other 2 terms (12 and 13) and answer the question. So Fact 1 is SUFFICIENT

With this same approach, we can deal with Fact 2.

The key to tackling most GMAT questions is to be comfortable breaking the prompt into logical pieces. Don't try to do every step at once and don't try to do work in your head. Think about what the information means, take the proper notes and be prepared to "play around" with a question if you're immediately certain about how to handle it.

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Re: What is the average (arithmetic mean) of eleven consecutive [#permalink]

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29 Jun 2016, 03:10
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wow such complex explanations for such a simple problem?

given :

11 consec integers

let them be x,x+1,x+2,...,x+10

Q: what is their mean?

mean is (11x+55)/11 = x+5.
Q becomes what is x+5

1) mean first 9 is 7.

so (9x+36)/9 = x+4 = 7 , so x+5 =8 ,--> sufficient A or D

2) mean of last 9 is 9.

so (9x+54)/9 = x+6= ---> x+5=8, sufficient . so D

D
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Re: What is the average (arithmetic mean) of eleven consecutive [#permalink]

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15 Jul 2016, 00:10
zz0vlb wrote:
What is the average (arithmetic mean) of eleven consecutive integers?

(1) The average of the first nine integers is 7
(2) The average of the last nine integers is 9

For odd number of consecutive integer the mean and median both is the middle value. Use this property to solve th question
(1) The average of the first nine integers is 7
7 will be the middle value; there will be 4 consecutive integers to the left and also to the right of 7
we will have {3,4,5,6,7,8,9,10,11}
now we can add last two consecutive integer after 11, they will be 12,13
our new set will become = {3,4,5,6,7,8,9,10,11,12,13}
again since the number of total elements in the set is odd, Mean will simply be the middle value = 8
SUFFICIENT

(2) The average of the last nine integers is 9
Again number of element in the set are odd, 9 will be the middle value; 4 consecutive integers will lie to its left and right
Middle value will be
{5,6,7,8,9,10,11,12,13}
Add 3,4 at the start of the set
new set = {3,4,5,6,7,8,9,10,11,12,13}
Mean will be 8
Sufficient

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Re: What is the average (arithmetic mean) of eleven consecutive [#permalink]

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20 Dec 2016, 17:37
Nice Official Question>
Here is my solution to this one =>
Set of consecutive integers =>
N
N+1
N+2
.
.
.
N+10

AP series with D=2
Hence Mean = Median = Average of the first and the last terms= N+5

So we just ned the value of N

Statement 1
N
N+1
.
.
N+8

Mean => N+4=7
Hence N+5=> 8
So the mean of the original set will be 8
Hence Sufficient

Statement 2-->
p
p+2
p+3
.
.
p+8
Mean = p+4=9
p=5
Hene p+8=>13
So N+10=>13
Hence N+5=>8
Hence the mean of the original data set must be 8

Hence Sufficient

Hence D

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Re: What is the average (arithmetic mean) of eleven consecutive [#permalink]

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26 Jan 2017, 05:52
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The key is that the integers are consecutive. So, if we can determine any one of the 11 (and know where it falls), we can answer the question.

(1) The average of the first 9 consecutive integers is 7.

We know that avg = sum of terms / # of terms.

So, 7 = sum of terms/9
sum of terms = 63.

Well, there's only going to be one set of 9 consecutive integers that add up to 63. If we can determine the first 9, we can certainly determine the last 2: sufficient.

(2) The average of the last 9 terms is 9.

Exact same reasoning as (1): sufficient.

Both (1) and (2) are sufficient: choose (D).
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Re: What is the average (arithmetic mean) of eleven consecutive [#permalink]

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13 Mar 2017, 05:39
The trick here is to catch the phase "11 consecutive integers". We know that an odd set of consecutive integers have the same median and mean {i.e. set 1,2,3 has a median and mean of 2}.

Based on this we can say the same for the statements:

s1) Represent the first 9 integers as: A+B+C+D+E+F+G+H+I. If the mean of this set is 7 then the median is also 7 so we found that the 7th number in the total set. We know they are consecutive and therefore we could count forward and backward to get the unknown numbers. Therefore statement is sufficient.

s2) Same concept as above. Statement is sufficient.
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Re: What is the average (arithmetic mean) of eleven consecutive [#permalink]

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24 Mar 2018, 07:32
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zz0vlb wrote:
What is the average (arithmetic mean) of eleven consecutive integers?

(1) The average of the first nine integers is 7
(2) The average of the last nine integers is 9

There's a nice rule that says, "In a set where the numbers are equally spaced, the mean will equal the median."

Since the consecutive integers are equally-spaced, their mean and median will be equal.

Target question: What is the average of eleven consecutive integers?

Statement 1: The average of the first nine integers is 7.
This also tells us that the MEDIAN of the first nine integers is 7.
In other words, the MIDDLEMOST value is 7.
This means, the first nine integers are 3, 4, 5, 6, 7, 8, 9, 10, 11
So, ALL 11 integers must be 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
Since we've identified all 11 integers, we can DEFINITELY find their average.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The average of the last night integers is 9
This also tells us that the MEDIAN of the last nine integers is 9.
In other words, the MIDDLEMOST value is 9.
This means, the last nine integers are 5, 6, 7, 8, 9, 10, 11, 12, 13
So, ALL 11 integers must be 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
Since we've identified all 11 integers, we can DEFINITELY find their average.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

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# Integration of secant

\begin{align} \int \sec x \, dx &= \int \cos x \left( \frac{1}{\cos^2x} \right) \, dx \\ &= \int \cos x \left( \frac{1}{1-\sin^2x} \right) \, dx \\ & = \int\cos x\cdot\frac{1}{1-\frac{1-\cos2x}{2}} \, dx \\ &= \int \cos x \cdot\frac{2}{1+\cos2x} \, dx \end{align}

I am stuck in here. Any help to integrate secant?

## 5 Answers

\begin{align*}\int\sec x\,\mathrm dx&=\int\frac1{\cos x}\,\mathrm dx\\&=\int\frac{\cos x}{\cos^2x}\,\mathrm dx\\&=\int\frac{\cos x}{1-\sin^2x}\,\mathrm dx.\end{align*} Now, doing $$\sin x=t$$ and $$\cos x\,\mathrm dx=\mathrm dt$$, you get $$\displaystyle\int\frac{\mathrm dt}{1-t^2}$$. But\begin{align*}\int\frac{\mathrm dt}{1-t^2}&=\frac12\int\frac1{1-t}+\frac1{1+t}\,\mathrm dt\\&=\frac12\left(-\log|1-t|+\log|1+t|\right)\\&=\frac12\log\left|\frac{1+t}{1-t}\right|\\&=\frac12\log\left|\frac{(1+t)^2}{1-t^2}\right|\\&=\log\left|\frac{1+t}{\sqrt{1-t^2}}\right|\\&=\log\left|\frac{1+\sin x}{\sqrt{1-\sin^2x}}\right|\\&=\log\left|\frac1{\cos x}+\frac{\sin x}{\cos x}\right|\\&=\log|\sec x+\tan x|.\end{align*}

• What a tricky..! Thx – Beverlie Jun 29 '17 at 15:25
• I've been reading about the early history of calculus. For a long period in the 17th century this was a significant unsolved problem. – DanielWainfleet Aug 3 '17 at 17:56
• @DanielWainfleet I think I read something about that in the historical notes of Spivak's Calculus. – José Carlos Santos Aug 3 '17 at 17:59

An alternative method: The trick here is to multiply $\sec{x}$ by $\dfrac{\tan{x}+\sec{x}}{\tan{x}+\sec{x}}$, then substitute $u=\tan{x}+\sec{x}$ and $du=(\sec^2{x}+\tan{x}\sec{x})~dx$:

$$\int \sec{x}~dx=\int \sec{x}\cdot \frac{\tan{x}+\sec{x}}{\tan{x}+\sec{x}}~dx=\int \frac{\sec{x}\tan{x}+\sec^2{x}}{\tan{x}+\sec{x}}~dx=\int \frac{1}{u}~du=\cdots$$

Not obvious, though it is efficient.

After $\int \cos x \left(\frac{1}{1-\sin^2x}\right)dx$ use the transformation $z = \sin x$ and $dz = \cos x \, dx$.

Edit:

$$\int\frac{1}{1-u^2}\,du = \frac{1}{2}\int\frac{(1+u)+(1-u)}{(1+u)(1-u)} = \frac{1}{2} \int \frac{1}{1+u} + \frac{1}{1-u}\,du$$

And use, $\int \frac{1}{u}\,du = \ln|u|$

• I'd got $\int \frac{1}{1-u^2}du$ what would be the next step? – Beverlie Jun 29 '17 at 15:22
• Use partial fraction method as in my edit. – Dhruv Kohli - expiTTp1z0 Jun 29 '17 at 15:26

Although the integral can be evaluated in a straightforward way using real analysis, I thought it might be instructive to present an approach based on complex analysis. To that end, we now proceed.

We use Euler's Formula, $e^{ix}=\cos(x)+i\sin(x)$, to write $\displaystyle \sec(x)=\frac2{e^{ix}+e^{-ix}}=\frac{2e^{ix}}{1+e^{i2x}}$. Then, we have

\begin{align} \int \sec(x)\,dx&=\int \frac2{e^{ix}+e^{-ix}}\\\\ &=\int \frac{2e^{ix}}{1+e^{i2x}}\,dx \\\\ &=-i2 \int \frac{1}{1+(e^{ix})^2}\,d(e^{ix})\\\\ &=-i2 \arctan(e^{ix})+C\tag 1\\\\ &=\log\left(\frac{1-ie^{ix}}{1+ie^{ix}}\right)+C\tag2\\\\ &=\log\left(-i\left(\frac{1+\sin(x)}{i\cos(x)}\right)\right)+C\tag3\\\\ &=\log(\sec(x)+\tan(x))+C'\tag4 \end{align}

NOTES:

In going from $(1)$ to $(2)$, we used the identity $\arctan(z)=i2\log\left(\frac{1-iz}{1+iz}\right)$

In going from $(2)$ to $(3)$, we multiplied the numerator and denominator of the argument of the logarithm function by $1-ie^{ix}$. Then, we used

$$\frac{1-ie^{ix}}{1+ie^{ix}}=\frac{-i2\cos(x)}{2(1-\sin(x))}=-i\frac{1+\sin(x)}{\cos(x)}$$

Finally, in going from $(3)$ to $(4)$, we absorbed the term $\log(-i)$ into the integration constant $C$ and labeled the new integration constant $C'=C+\log(-i)$.

Just to spell out Lord Shark the Unknown's suggestion, $$t=\tan\frac{x}{2}\implies\sec x=\frac{1+t^2}{1-t^2},\,dx=\frac{2dt}{1+t^2}\implies\int\sec xdx=\int\frac{2 dt}{1-t^2}.$$From that point on, the same partial-fractions treatment as in multiple other answers can be used. Admittedly the expression thus obtained for the antiderivative is $$\ln\left|\frac{1+\tan\frac{x}{2}}{1-\tan\frac{x}{2}}\right|+C$$ instead of $$\ln\left|\frac{1+\sin x}{1-\sin x}\right|+C$$ or $$\ln|\sec x+\tan x|+C$$, but of course they're all the same thanks to suitable trigonometric identities (again, obtainable by writing things as functions of $$\tan\frac{x}{2}$$).


But you might be surprised because nature seems to favor a particular numbers like 1, 2, 3, 5, 8, 13, 21 and 34. Lucas Sequences The above work on the Fibonacci sequence can be generalized to discuss any difference equation of the form where and can be any real numbers. The Fibonacci sequence is a series where the next term is the sum of pervious two terms. Is there an easier way? Writing, the other root is, and the constants making are. The Explicit Formula for Fibonacci Sequence First, let's write out the recursive formula: a n + 2 = a n + 1 + a n a_{n+2}=a_{n+1}+a_n a n + 2 = a n + 1 + a n where a 1 = 1 , a 2 = 1 a_{ 1 }=1,\quad a_2=1 a 1 = 1 , a 2 = 1 You might think that any number is possible. How does this Fibonacci calculator work? It goes by the name of golden ratio, which deserves its own separate article.). They hold a special place in almost every mathematician's heart. Fibonacci numbers are one of the most captivating things in mathematics. Fibonacci Sequence is a wonderful series of numbers that could start with 0 or 1. For example, in the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13,... 2 is found by adding the two numbers before it, 1+1=2. The nth term of a Fibonacci sequence is found by adding up the two Fibonacci numbers before it. Try it again. If we have an infinite series, $$S = 1 + ax + (ax)^2 + (ax)^3 + \cdots,$$, with $|ax| < 1$, then its sum is given by, This means, if the sum of an infinite geometric series is finite, we can always have the following equality -, $$\frac{1}{1 - ax} = 1 + ax + (ax)^2 + (ax)^3 + \cdots = \sum_{n \ge 0} a^n x^n$$, Using this idea, we can write the expression of $F(x)$ as, $$F(x) = \frac{1}{(\alpha - \beta)}\left(\frac{1}{1-x\alpha} - \frac{1}{1-x\beta} \right) = \frac{1}{\sqrt{5}} \left(\sum_{n \ge 0 } x^n\alpha^n - \sum_{n \ge 0 } x^n \beta^n \right)$$, Recalling the original definition of $F(x)$, we can finally write the following equality, $$F(x) = \sum_{n \ge 0}F_n x^n = \frac{1}{\sqrt{5}} \left(\sum_{n \ge 0 } x^n\alpha^n - \sum_{n \ge 0 } x^n \beta^n \right),$$, and comparing the $n-$th terms on both sides, we get a nice result, $$F_n = \frac{1}{\sqrt{5}} \left(\alpha^n - \beta^n \right),$$, (This number $\alpha$ is also a very interesting number in itself. Yes, there is an exact formula for the n … Thus the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …. Throughout history, people have done a lot of research around these numbers, and as a result, quite a lot … . F(n) = F(n+2) - F(n+1) F(n-1) = F(n+1) - F(n) . In mathematical terms, the sequence F n of all Fibonacci numbers is defined by the recurrence relation. Yes, it is possible but there is an easy way to do it. So, with the help of Golden Ratio, we can find the Fibonacci numbers in the sequence. Fibonacci spiral is also considered as one of the approximates of the golden spiral. Here is a short list of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 Each number in the sequence is the sum of the two numbers before it We can try to derive a Fibonacci sequence formula by making some observations So, the sequence goes as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. Fibonacci omitted the first term (1) in Liber Abaci. Fibonacci sequence formula Golden ratio convergence A natural derivation of the Binet's Formula, the explicit equation for the Fibonacci Sequence. See more ideas about fibonacci, fibonacci spiral, fibonacci sequence. Follow me elsewhere: Twitter: https://twitter.com/RecurringRoot This will give you the second number in the sequence. There is a special relationship between the Golden Ratio and the Fibonacci Sequence:. Fibonacci number - elements of a numerical sequence in which the first two numbers are either 1 and 1, or 0 and 1, and each subsequent number is equal to the sum of the two previous numbers. Specifically, we have noted that the Fibonacci sequence is a linear recurrence relation — it can be viewed as repeatedly applying a linear map. Jacques Philippe Marie Binet was a French mathematician, physicist, and astronomer born in Rennes. So, for n>1, we have: f₀ = 0, f₁ = 1, This short project is an implementation of the formula in C. Binet's Formula . A Closed Form of the Fibonacci Sequence Fold Unfold. By the above formula, the Fibonacci number can be calculated in . Fibonacci initially came up with the sequence in order to model the population of rabbits. # first two terms n1, n2 = 0, 1 count = 0 # check if the number of terms is valid if nterms <= 0: print("Please enter a positive integer") elif nterms == 1: print("Fibonacci sequence upto",nterms,":") print(n1) else: print("Fibonacci sequence:") while count < nterms: print(n1) nth = n1 + n2 # update values n1 = n2 n2 = … The third number in the sequence is the first two numbers added together (0 + 1 = 1). The Fibonacci Sequence is one of the cornerstones of the math world. Fibonacci spiral is also considered as one of the approximates of the golden spiral. The answer key is below. Python Fibonacci Sequence: Iterative Approach. The Fibonacci sequence exhibits a certain numerical pattern which originated as the answer to an exercise in the first ever high school algebra text. If we make the replacement. Problems to be Submitted: Problem 10. Keywords and phrases: Generalized Fibonacci sequence, Binet’s formula. I know that the relationship is that the "sum of the squares of the first n terms is the nth term multiplied by the (nth+1) term", but I don't think that is worded right? Each number in the sequence is the sum of the two previous numbers. The first two numbers of the Fibonacci series are 0 and 1. Leonardo Fibonacci was one of the most influential mathematician of the middle ages because Hindu Arabic Numeral System which we still used today was popularized in the Western world through his book Liber Abaci or book of calculations. This sequence of Fibonacci numbers arises all over mathematics and also in nature. In reality, rabbits do not breed this… With this insight, we observed that the matrix of the linear map is non-diagonal, which makes repeated execution tedious; diagonal matrices, on the other hand, are easy to multiply. Assuming "Fibonacci sequence" is an integer sequence | Use as referring to a mathematical definition or referring to a type of number instead. The recurrence formula for these numbers is: F(0) = 0 F(1) = 1 F(n) = F(n − 1) + F(n − 2) n > 1 . I have been learning about the Fibonacci Numbers and I have been given the task to research on it. Get all the latest & greatest posts delivered straight to your inbox, © 2020 Physics Garage. The Fibonacci numbers, denoted fₙ, are the numbers that form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones. The first two numbers are defined to be 0, 1. where $n$ is a positive integer greater than $1$, $F_n$ is the $n-$th Fibonacci number with $F_0 = 0$ and $F_1=1$. I have been assigned to decribe the relationship between the photo (attached below). The Fibonacci numbers are generated by setting F 0 = 0, F 1 = 1, and then using the recursive formula. In mathematics, the Fibonacci numbers form a sequence defined recursively by: = {= = − + − > That is, after two starting values, each number is the sum of the two preceding numbers. Male or Female ? Each number is the product of the previous two numbers in the sequence. They hold a special place in almost every mathematician's heart. Generate Fibonacci sequence (Simple Method) In the Fibonacci sequence except for the first two terms of the sequence, every other term is the sum of the previous two terms. So, … Fibonacci formula: f … It may seem coincidence to you but it's actually forming a pattern - Fibonacci Sequence. Each number in the sequence is the sum of the two numbers that precede it. Add the first term (1) and 0. Binet's Formula is an explicit formula used to find the nth term of the Fibonacci sequence. Stay up to date! The Fibonacci sequence typically has first two terms equal to F₀ = 0 and F₁ = 1. Male Female Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation Elementary school/ Junior high-school student The answer comes out as a whole number, exactly equal to the addition of the previous two terms. Assuming "Fibonacci sequence" is an integer sequence | Use as referring to a mathematical definition or referring to a type of number instead. Unlike in an arithmetic sequence, you need to know at least two consecutive terms to figure out the rest of the sequence. The mathematical equation describing it is An+2= An+1 + An. Browse other questions tagged sequences-and-series fibonacci-numbers or ask your own question. . The first two numbers are defined to be 0, 1. This is the general form for the nth Fibonacci number. In this book, Fibonacci post and solve a problem involving the growth of population of rabbits based on idealized assumptions. The Golden Ratio formula is: F(n) = (x^n – (1-x)^n)/(x – (1-x)) where x = (1+sqrt 5)/2 ~ 1.618. To improve this 'Fibonacci sequence Calculator', please fill in questionnaire. The Explicit Formula for Fibonacci Sequence First, let's write out the recursive formula: a n + 2 = a n + 1 + a n a_{n+2}=a_{n+1}+a_n a n + 2 = a n + 1 + a n where a 1 = 1 , a 2 = 1 a_{ 1 }=1,\quad a_2=1 a 1 = 1 , a 2 = 1 A natural derivation of the Binet's Formula, the explicit equation for the Fibonacci Sequence. Where, φ is the Golden Ratio, which is approximately equal to the value 1.618. n is the nth term of the Fibonacci sequence To calculate each successive Fibonacci number in the Fibonacci series, use the formula where is th Fibonacci number in the sequence, and the first … Next, we multiply the last equation by $x_n$ to get, $$x^n \cdot F_{n+1} = x^n \cdot F_n + x^n \cdot F_{n-1},$$, $$\sum_{n \ge 1}x^n \cdot F_{n+1} = \sum_{n \ge 1} x^n \cdot F_n + \sum_{n \ge 1} x^n \cdot F_{n-1}$$, Let us first consider the left hand side -, $$\sum_{n \ge 1} x^n \cdot F_{n+1} = x \cdot F_2 + x^2 \cdot F_3 + \cdots$$, Now, we try to represent this expansion in terms of $F(x)$, by doing the following simple manipulations -, $$\frac{1}{x} \left( x^2 \cdot F_2 + x^3 \cdot F_3 + \cdots \right)$$, $$\frac{1}{x} \left(- x \cdot F_1 + x \cdot F_1 + x^2 \cdot F_2 + x^3 \cdot F_3 + \cdots \right)$$, Using the definition of $F(x)$, this expression can now be written as, $$\frac{1}{x} \left(- x \cdot F_1 + F(x)\right)$$, Therefore, using the fact that $F_1=1$, we can write the entire left hand side as, $$\sum_{n \ge 1} x^n \cdot F_{n+1} = x \cdot F_2 + x^2 \cdot F_3 + \cdots = \frac{F(x) - x}{x}$$, $$\sum_{n \ge 1}x^n \cdot F_n + \sum_{n \ge 1} x^n \cdot F_{n-1}.$$, $$\left( x \cdot F_1 + x^2 \cdot F_2 + \cdots \right ) + \left( x^2 \cdot F_1 + x^3 \cdot F_2 + \cdots \right)$$. This equation calculates numbers in the Fibonacci sequence (Fn) by adding together the previous number in the series (Fn-1) with the number previous to that (Fn-2). x(n-2) is the term before the last one. Male or Female ? The powers of phi are the negative powers of Phi. Fibonacci Number Formula. . Alternatively, you can choose F₁ = 1 and F₂ = 1 as the sequence starters. Generalized Fibonacci sequence is defined by recurrence relation F pF qF k with k k k t 12 F a F b 01,2, Male Female Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation Elementary school/ Junior high-school student By the above formula, the Fibonacci number can be calculated in . Computing Fibonacci number by exponentiation. Table of Contents. If F(n) represents the nth Fibonacci number, then: F(n) = (a^n - b^n)/(a - b) where a and b are the two roots of the quadratic equation x^2-x-1 = 0. Instead, it would be nice if a closed form formula for the sequence of numbers in the Fibonacci sequence existed. We can also use the derived formula below. For example, in the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13,... 2 is found by adding the two numbers before it, 1+1=2. Francis Niño Moncada on October 01, 2020: Jomar Kristoffer Besayte on October 01, 2020: Mary Kris Banaynal on September 22, 2020: Ace Victor A. Acena on September 22, 2020: Andrea Nicole Villa on September 22, 2020: Claudette Marie Bonagua on September 22, 2020: Shaira A. Golondrina on September 22, 2020: Diana Rose A. Orillana on September 22, 2020: Luis Gabriel Alidogan on September 22, 2020: Grace Ann G. Mohametano on September 22, 2020. Instead, it would be nice if a closed form formula for the sequence of numbers in the Fibonacci sequence existed. A sequence derived from this equation is often called a Lucas sequence, named for French mathematician Edouard Lucas. The rule for calculating the next number in the sequence is: x(n) = x(n-1) + x(n-2) x(n) is the next number in the sequence. to get the rest. Remember, to find any given number in the Fibonacci sequence, you simply add the two previous numbers in the sequence. The first two terms of the Fibonacci sequence is 0 followed by 1. To calculate each successive Fibonacci number in the Fibonacci series, use the formula where is th Fibonacci number in the sequence, and the first … Fibonacci initially came up with the sequence in order to model the population of rabbits. In his memoir in the theory of conjugate axis and the moment of inertia of bodies, he enumerated the principle which is known now as Binet's Theorem. The Fibonacci sequence is a series where the next term is the sum of pervious two terms. A Fibonacci spiral having an initial radius of 1 has a polar equation similar to that of other logarithmic spirals . F n – 1 and F n – 2 are the (n-1) th and (n – 2) th terms respectively Also Check: Fibonacci Calculator. Solution for 88. $$0, 1, 1, 2, 3, 5, 8, 13 ,21, 34, 55, \cdots$$, Any number in this sequence is the sum of the previous two numbers, and this pattern is mathematically written as. Fibonacci Sequence. This pattern turned out to have an interest and … The Fibonacci Sequence is a series of numbers. The characteristic equation is, with roots. Although Fibonacci only gave the sequence, he obviously knew that the nth number of his sequence was the sum of the two previous numbers (Scotta and Marketos). # Program to display the Fibonacci sequence up to n-th term nterms = int(input("How many terms? ")) Abstract. A Fibonacci spiral having an initial radius of 1 has a polar equation similar to that of other logarithmic spirals . The Fibonacci series is a very famous series in mathematics. A Closed Form of the Fibonacci Sequence Fold Unfold. Example 2: Find the 25th term of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... Answer: Since you're looking for the 25th term, n = 25. Fibonacci number is defined by: Obviously, Fibonacci sequence is a difference equation (in above example) and it could be written in: Matrix Form. Computing Fibonacci number by exponentiation. In this tutorial I will show you how to generate the Fibonacci sequence in Python using a few methods. You can use the Binet's formula in in finding the nth term of a Fibonacci sequence without the other terms. The nth term of a Fibonacci sequence is found by adding up the two Fibonacci numbers before it. Now, this expression is fairly easy to understand and quite sufficient to produce any Fibonacci number by plugging the required value of $n$. By taking out a factor of $x$ from the second expansion, we get, $$\left( x \cdot F_1 + x^2 \cdot F_2 + \cdots \right ) + x \left( x \cdot F_1 + x^2 \cdot F_2 + \cdots \right).$$, Using the definition of $F(x)$, this can finally be written as. This pattern turned out to have an interest and … Our job is to find an explicit form of the function, $F(x)$, such that the coefficients, $F_n$ are the Fibonacci numbers. I know that the relationship is that the "sum of the squares of the first n terms is the nth term multiplied by the (nth+1) term", but I don't think that is worded right? If you got 4 correct answers: You made it! ( Using power of the matrix {{1,1},{1,0}} ) This another O(n) which relies on the fact that if we n times … If you got between 0 and 1 correct answer: You can do it next time. It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre. Fibonacci sequence is a sequence of numbers, where each number is the sum of the 2 previous numbers, except the first two numbers that are 0 and 1. The equation is a variation on Pell's, in that x^2 - ny^2 = +/- 4 instead of 1. Fibonacci number is defined by: Obviously, Fibonacci sequence is a difference equation (in above example) and it could be written in: Matrix Form. The problem yields the ‘Fibonacci sequence’: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 . Let us define a function $F(x)$, such that it can be expanded in a power series like this, $$F(x) = \sum_{n \ge 0}x^n F_n = x \cdot F_1 + x^2 \cdot F_2 + \cdots$$. Following the same pattern, 3 is found by adding 1 and 2, 5 is found by adding 2 and 3 and so on. So to calculate the 100th Fibonacci number, for instance, we need to compute all the 99 values before it first - quite a task, even with a calculator! F n = F n-1 + F n-2. The Fibonacci formula is used to generate Fibonacci in a recursive sequence. Fibonacci Sequence. Therefore, by equating the left and the right hand sides, the original formula can be re-written in terms of $F(x)$ as, $$\frac{F(x) - x}{x} = F(x) + xF(x) ~~ \Longrightarrow ~~ F(x) = \frac{x}{1-x-x^2}$$, Let us now simplify this expression a bit more. Mar 12, 2018 - Explore Kantilal Parshotam's board "Fibonacci formula" on Pinterest. The formula to calculate the Fibonacci numbers using the Golden Ratio is: X n = [φ n – (1-φ) n]/√5. Fibonacci Sequence is popularized in Europe by Leonardo of Pisa, famously known as "Leonardo Fibonacci".Leonardo Fibonacci was one of the most influential mathematician of the middle ages because Hindu Arabic Numeral System which we still used today was popularized in the Western world through his book Liber Abaci or book of calculations. If you got between 2 and 3 correct answers: Maybe you just need more practice. If we expand the by taking in above example, then. From this we find the formula, valid for all, and one desired continuous extension is clearly the real part Derivation of Fibonacci sequence . Fibonacci Sequence. THE FIBONACCI SEQUENCE, SPIRALS AND THE GOLDEN MEAN. It is not hard to imagine that if we need a number that is far ahead into the sequence, we will have to do a lot of "back" calculations, which might be tedious. However, if I wanted the 100th term of this sequence, it would take lots of intermediate calculations with the recursive formula to get a result. . Fibonacci Formula. Observe the following Fibonacci series: Derivation of Fibonacci sequence . The standard formula for the Fibonacci numbers is due to a French mathematician named Binet. Example 1: Find the 10th term of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... Answer: Since you're looking for the 10th term, n = 10. In order to make use of this function, first we have to rearrange the original formula. The sequence starts like this: 0, 1, 1, 2, 3, 4, 8, 13, 21, 34 (Issues regarding the convergence and uniqueness of the series are beyond the scope of the article). This sequence of Fibonacci numbers arises all over mathematics and also in nature. Number Theory > Special Numbers > Fibonacci Numbers > Binet's Fibonacci Number Formula Binet's formula is a special case of the Binet form with , corresponding to the th Fibonacci … . To improve this 'Fibonacci sequence Calculator', please fill in questionnaire. And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio: x n = φn − (1−φ)n √5. In this article, we are going to discuss another formula to obtain any Fibonacci number in the sequence, which might (arguably) be easier to work with. You can calculate the Fibonacci Sequence by starting with 0 and 1 and adding the previous two numbers, but Binet's Formula can be used to calculate directly any term of the sequence. Subscribe to the newsletter to receive more stories mailed directly to your inbox, The methods of finding roots of a quadratic equations are quite easy and are very well understood. Using The Golden Ratio to Calculate Fibonacci Numbers. In reality, rabbits do not breed this… The first two terms of the Fibonacci sequence is 0 followed by 1. The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation (1) Fibonacci Series Formula. Forty years ago I discovered that the Fibonacci Sequence (1, 1, 2, 3, 5, 8, etc) can be generated from the second degree Diophantine equation 5k^2 -/+ 4 = m^2 where the -,+ is taken alternately. The Fibonacci sequence was defined in Section 11.1 by the equations fi = 1, f2= 1, fn= fn=1 + fn-2 n> 3 %3D %3D Show that each of the following… Get the best viral stories straight into your inbox! He died in Paris, France in 1856. So, for n>1, we have: f₀ = 0, f₁ = 1, Let’s start by talking about the iterative approach to implementing the Fibonacci series. To create the sequence, you should think of 0 … Fibonacci number - elements of a numerical sequence in which the first two numbers are either 1 and 1, or 0 and 1, and each subsequent number is equal to the sum of the two previous numbers. The authors would like to thank Prof. Ayman Badawi for his fruitful suggestions. The Fibonacci sequence is one of the most famous formulas in mathematics. 1 Binet's Formula for the nth Fibonacci number We have only defined the nth Fibonacci number in terms of the two before it: the n-th Fibonacci number is the sum of the (n-1)th and the (n-2)th. In this paper, we present properties of Generalized Fibonacci sequences. where: a = (F₁ - F₀ψ) / √5 b = (φF₀ - F₁) / √5 F₀ is the first term of the sequence, F₁ is the second term of the sequence. But what if you are asked to find the 100th term of a Fibonacci sequence, are you going to add the Fibonacci numbers consecutively until you get the 100th term? THE FIBONACCI SEQUENCE, SPIRALS AND THE GOLDEN MEAN. Fibonacci sequence equation. If at all, its only drawback is that, if we want to know a particular number, $F_n$ in the sequence, we need two numbers $F_{n-1}$ and $F_{n-2}$ that came before it; that's just how this formula works. F n = n th term of the series. Another way to write the equation is: Therefore, phi = 0.618 and 1/Phi. There are all kinds of approaches available, like, Ptolemy was an ancient astronomer, geographer, and mathematician who lived from (c. AD 100 – c. 170). In the case of the Fibonacci sequence, the recurrence is, with initial conditions. We can see from the following table, that by plugging the values of $n$, we can directly find all Fibonacci numbers! I have been assigned to decribe the relationship between the photo (attached below). For each question, choose the best answer. Featured on Meta “Question closed” notifications experiment results and graduation Definition The Fibonacci sequence begins with the numbers 0 and 1. x(n-1) is the previous term. . In mathematics, the Fibonacci sequence is defined as a number sequence having the particularity that the first two numbers are 0 and 1, and that each subsequent number is obtained by the sum of the previous two terms. Fibonacci Sequence is popularized in Europe by Leonardo of Pisa, famously known as "Leonardo Fibonacci". Scope of the formula in C. Binet 's formula, the recurrence is, and then using the recursive.! Liber Abaci is so named because it was derived by mathematician Jacques Philippe Marie Binet, though was. N = n th term of the Golden MEAN the Binet 's formula in C. Binet 's formula C.... Numbers arises all over mathematics and also in nature he made significant contributions to number theory and the sequence... 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First two numbers are generated by adding up the two Fibonacci numbers before it famous formulas in mathematics phi... Delivered straight to your inbox viral stories straight into your inbox, © 2020 Physics Garage & greatest posts straight.
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