Problems

2002 Paper 1 Q10

D: 1516.0 B: 1470.2

momentum and collisions elastic collision recurrence relation bicycle pump particle bouncing telescoping product relative velocity sequences

A bicycle pump consists of a cylinder and a piston. The piston is pushed in with steady speed~$u$. A particle of air moves to and fro between the piston and the end of the cylinder, colliding perfectly elastically with the piston and the end of the cylinder, and always moving parallel with the axis of the cylinder. Initially, the particle is moving towards the piston at speed $v$. Show that the speed, $v_n$, of the particle just after the $n$th collision with the piston is given by $v_n=v+2nu$. Let $d_n$ be the distance between the piston and the end of the cylinder at the $n$th collision, and let $t_n$ be the time between the $n$th and $(n+1)$th collisions. Express $d_n - d_{n+1}$ in terms of $u$ and $t_n$, and show that \[ d_{n+1} = \frac{v+(2n-1)u}{v+(2n+1)u} \, d_n \;. \] Express $d_n$ in terms of $d_1$, $u$, $v$ and $n$. In the case $v=u$, show that $ut_n = \displaystyle \frac {d_1} {n(n+1)}$. %%%%%Verify that $\sum\limits_1^\infty t_n = d/u$.

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2002 Paper 2 Q5

D: 1600.0 B: 1495.1

proof by induction recurrence relation sequences quadratic logistic map fixed points periodic orbits inequality

The numbers $x_n$, where $n=0$, $1$, $2$, $\ldots$ , satisfy \[ x_{n+1} = kx_n(1-x_n) \;. \]

Prove that, if $0 < k < 4$ and $0 < x_0 < 1$, then $0 < x_n < 1$ for all $n\,$.
Given that $x_0=x_1=x_2 = \cdots =a\,$, with $a\ne0$ and $a\ne1$, find $k$ in terms of $a\,$.
Given instead that $x_0=x_2=x_4 = \cdots = a\,$, with $a\ne0$ and $a\ne1$, show that $ab^3 -b^2 +(1-a)=0$, where $b=k(1-a)\,$. Given, in addition, that $x_1 \ne a$, find the possible values of $k$ in terms of $a\,$.

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2002 Paper 3 Q12

D: 1700.0 B: 1502.1

probability conditional probability expectation recurrence relation geometric series biased coin repeated trials successive events

In a game, a player tosses a biased coin repeatedly until two successive tails occur, when the game terminates. For each head which occurs the player wins $\pounds 1$. If $E$ is the expected number of tosses of the coin in the course of a game, and $p$ is the probability of a head, explain why \[ E = p \l 1 + E \r + \l 1 - p \r p \l 2 + E \r + 2 \l 1 - p \r ^2\,, \] and hence determine $E$ in terms of $p$. Find also, in terms of $p$, the expected winnings in the course of a game. A second game is played, with the same rules, except that the player continues to toss the coin until $r$ successive tails occur. Show that the expected number of tosses in the course of a game is given by the expression $\displaystyle {1 - q^r \over p q^r}\,$, where $q = 1 - p$.

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2001 Paper 3 Q1

D: 1700.0 B: 1500.0

Taylor series Maclaurin series proof by induction implicit differentiation Leibniz rule recurrence relation logarithmic function inverse hyperbolic function

Given that $y = \ln ( x + \sqrt{x^2 + 1})$, show that $ \displaystyle \frac{\d y}{\d x} = \frac1 {\sqrt{x^2 + 1} }\;$. Prove by induction that, for $n \ge 0\,$, \[ \l x^2 + 1 \r y^{\l n + 2 \r} + \l 2n + 1 \r x y^{\l n + 1 \r} + n^2 y^{\l n \r} = 0\;, \] where $\displaystyle y^{(n)} = \frac{\d^n y}{\d x^n}$ and $y^{(0)} =y\,$. Using this result in the case $x = 0\,$, or otherwise, show that the Maclaurin series for $y$ begins \[ x - {x^3 \over 6} +{3 x^5 \over 40} \] and find the next non-zero term.

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2001 Paper 3 Q13

D: 1700.0 B: 1500.0

probability conditional probability sequential game fair coin recurrence relation Markov chain non-transitive game theory

In a game for two players, a fair coin is tossed repeatedly. Each player is assigned a sequence of heads and tails and the player whose sequence appears first wins. Four players, $A$, $B$, $C$ and $D$ take turns to play the game. Each time they play, $A$ is assigned the sequence TTH (i.e.~Tail then Tail then Head), $B$ is assigned THH, $C$ is assigned HHT and $D$ is assigned~HTT.

$A$ and $B$ play the game. Let $p_{\mathstrut\mbox{\tiny HH}}$, $p_{\mathstrut\mbox{\tiny HT}}$, $p_{\mathstrut\mbox{\tiny TH}}$ and $p_{\mathstrut\mbox{\tiny TT}}$ be the probabilities of $A$ winning the game given that the first two tosses of the coin show HH, HT, TH and TT, respectively. Explain why $p_{\mathstrut\mbox{\tiny TT}} = 1\,$, and why $p_{\mathstrut\mbox{\tiny HT}} = {1 \over 2} \, p_{\mathstrut\mbox{\tiny TH}} + {1\over 2} \, p_{\mathstrut\mbox{\tiny TT}}\,$. Show that $p_{\mathstrut\mbox{\tiny HH}} = p_{\mathstrut\mbox{\tiny HT}} = {2 \over 3}$ and that $p_{\mathstrut\mbox{\tiny TH}} = {1\over 3}\,$. Deduce that the probability that A wins the game is ${2\over 3}\,$.
$B$ and $C$ play the game. Find the probability that $B$ wins.
Show that if $C$ plays $D$, then $C$ is more likely to win than $D$, but that if $D$ plays $A$, then $D$ is more likely to win than $A$.

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2000 Paper 3 Q8

D: 1700.0 B: 1484.0

sequences recurrence relation proof by induction auxiliary equation golden ratio nonlinear recurrence closed form

The sequence $a_n$ is defined by $a_0 = 1$ , $a_1 = 1$ , and $$ a_n = {1 + a_{n - 1}^2 \over a_{n - 2} } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ( n \ge 2 ) . $$ Prove by induction that $$ a_n = 3 a_{n - 1} - a_{n - 2} \ \ \ \ \ \ \ \ \ \ \ ( n \ge2 ) . $$ Hence show that $$ a_n = {\alpha^{2 n - 1} + \alpha^{- ( 2 n - 1 )} \over \sqrt 5} \ \ \ \ \ \ (n\ge1), $$ where $\displaystyle{\alpha = {1 + \sqrt 5 \over 2}}$.

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1999 Paper 1 Q3

D: 1500.0 B: 1500.0

inequalities sequences recurrence relation proof by contradiction quadratics cyclic system golden ratio

The $n$ positive numbers $x_{1},x_{2},\dots,x_{n}$, where $n\ge3$, satisfy $$ x_{1}=1+\frac{1}{x_{2}}\, ,\ \ \ x_{2}=1+\frac{1}{x_{3}}\, , \ \ \ \dots\; , \ \ \ x_{n-1}=1+\frac{1}{x_{n}}\, , $$ and also $$ \ x_{n}=1+\frac{1}{x_{1}}\, . $$ Show that

$x_{1},x_{2},\dots,x_{n}>1$,
${\displaystyle x_{1}-x_{2}=-\frac{x_{2}-x_{3}}{x_{2}x_{3}}}$,
$x_{1}=x_{2}=\cdots=x_{n}$.

Hence find the value of $x_1$.

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1999 Paper 3 Q5

D: 1700.0 B: 1516.0

sequences recurrence relation Fibonacci sequence proof by induction algebraic identity summation of squares

The sequence $u_0$, $u_1$, $u_2$, ... is defined by $$ u_0=1,\hspace{0.2in} u_1=1, \quad u_{n+1}=u_n+u_{n-1} \hspace{0.2in}{\rm for}\hspace{0.1in}n \ge 1\,. $$ Prove that $$ u^2_{n+2} + u^2_{n-1} = 2( u^2_{n+1} + u^2_n ) \,. $$ Using induction, or otherwise, prove the following result: \[ u_{2n} = u^2_n + u^2_{n-1} \quad \mbox{ and }\quad u_{2n+1} = u^2_{n+1} - u^2_{n-1} \] for any positive integer $n$.

Solution: Claim: $u^2_{n+2} + u^2_{n-1} = 2( u^2_{n+1} + u^2_n )$ Proof: (By Induction). (Base Case): $n = 1$ \begin{align*} && LHS &= u_{n+2}^2 + u_{n-1}^2 \\ &&&= u_3^2 + u_0^2 \\ &&&= 3^2 + 1^2 = 10\\ && RHS &= 2(u_{n+1}^2+u_n^2) \\ &&&= 2(2^2 + 1^2) \\ &&&= 10 \end{align*} Therefore the base case is true. (Inductive hypothesis) Suppose $u^2_{n+2} + u^2_{n-1} = 2( u^2_{n+1} + u^2_n )$ is true for some $n = k$, ie $u^2_{k+2} + u^2_{k-1} = 2( u^2_{k+1} + u^2_k )$, the consider $n = k+1$ \begin{align*} && LHS &= u_{k+1+2}^2 + u_{k+1-1}^2 \\ &&&= (u_{k+1}+u_{k+2})^2+u_k^2 \\ &&&= u_{k+2}^2+u_{k+1}^2+ u_k^2 + 2u_{k+1}u_{k+2} \\ &&&= u_{k+2}^2+u_{k+1}^2+ u_k^2 + 2u_{k+1}(u_{k+1}+u_k) \\ &&&= u_{k+2}^2 + u_{k+1}^2+u_k^2+2u_{k+1}^2+2u_{k+1}u_k \\ &&&= u_{k+1}^2+2u_{k+1}^2+ u_{k+1}^2+u_k^2+2u_{k+1}u_k \\ &&&= u_{k+2}^2+2u_{k+1}^2+ (u_{k+1}+u_k)^2 \\ &&&= u_{k+2}^2+2u_{k+1}^2+ u_{k+2}^2 \\ &&&=2(u_{k+2}^2+u_{k+1}^2) \\ &&&= RHS \end{align*} Therefore it is true for $n = k+1$. Therefore by the principle of mathematical induction it is true for all $n \geq 1$ Claim: $ u_{2n} = u^2_n + u^2_{n-1} \quad \mbox{ and }\quad u_{2n+1} = u^2_{n+1} - u^2_{n-1} $ Proof: Notice that $\begin{pmatrix}u_{n+1} \\ u_n \end{pmatrix}= \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n \begin{pmatrix}1 \\1 \end{pmatrix}$, in particular \begin{align*} && \begin{pmatrix}u_{n}& u_{n-1} \\ u_{n-1} & u_{n-2} \end{pmatrix}&= \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^{n} \\ \Rightarrow && \begin{pmatrix}u_{2n}& u_{2n-1} \\ u_{2n-1} & u_{2n-2} \end{pmatrix}&= \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^{2n} \\ &&&= \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^{n} \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^{n} \\ &&&=\begin{pmatrix}u_{n}& u_{n-1} \\ u_{n-1} & u_{n-2} \end{pmatrix}\begin{pmatrix}u_{n}& u_{n-1} \\ u_{n-1} & u_{n-2} \end{pmatrix}\\ &&&= \begin{pmatrix}u_{n}^2+u_{n-1}^2& u_{n-1}(u_n+u_{n-2}) \\ u_{n-1}(u_n+u_{n-2}) & u_{n-1}^2+u_{n-2}^2 \end{pmatrix} \end{align*} Therefore $u_{2n} = u_{n}^2+u_{n-1}^2$ and $u_{2n+1} = u_n(u_{n+1}+u_{n-1}) =(u_{n+1}-u_{n-1})(u_{n+1}-u_{n-1}) = u_{n+1}^2-u_{n-1}^2$

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1998 Paper 1 Q6

D: 1500.0 B: 1500.0

trigonometry recurrence relation half-angle formula product of cosines sequences proof by induction cos x arithmetic-geometric mean

Let $a_{1}=\cos x$ with $0 < x < \pi/2$ and let $b_{1}=1$. Given that \begin{eqnarray*} a_{n+1}&=&{\textstyle \frac{1}{2}}(a_{n}+b_{n}),\\[2mm] b_{n+1}&=&(a_{n+1}b_{n})^{1/2}, \end{eqnarray*} find $a_{2}$ and $b_{2}$ and show that \[a_{3}=\cos\frac{x}{2}\cos^{2}\frac{x}{4} \ \quad\mbox{and}\quad \ b_{3}=\cos\frac{x}{2}\cos\frac{x}{4}.\] Guess general expressions for $a_{n}$ and $b_{n}$ (for $n\ge2$) as products of cosines and verify that they satisfy the given equations.

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1998 Paper 2 Q3

D: 1600.0 B: 1516.0

sequences series partial fractions telescoping series convergence recurrence relation summation product telescoping

Show that the sum $S_N$ of the first $N$ terms of the series $$\frac{1}{1\cdot2\cdot3}+\frac{3}{\cdot3\cdot4}+\frac{5}{3\cdot4\cdot5}+\cdots +\frac{2n-1}{n(n+1)(n+2)}+\cdots$$ is $${1\over2}\left({3\over2}+{1\over N+1}-{5\over N+2}\right).$$ What is the limit of $S_N$ as $N\to\infty$? The numbers $a_n$ are such that $$\frac{a_n}{a_{n-1}}=\frac{(n-1)(2n-1)}{(n+2)(2n-3)}.$$ Find an expression for $a_n/a_1$ and hence, or otherwise, evaluate $\sum\limits_{n=1}^\infty a_n$ when $\displaystyle a_1=\frac{2}{9}\;$.

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1998 Paper 2 Q4

D: 1600.0 B: 1470.2

integration reduction formula trigonometric telescoping sum product-to-sum formula definite integral recurrence relation

The integral $I_n$ is defined by $$I_n=\int_0^\pi(\pi/2-x)\sin(nx+x/2)\,{\rm cosec}\,(x/2)\,\d x,$$ where $n$ is a positive integer. Evaluate $I_n-I_{n-1}$, and hence evaluate $I_n$ leaving your answer in the form of a sum.

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1998 Paper 3 Q2

D: 1700.0 B: 1500.0

integration Beta function integration by parts substitution recurrence relation factorial symmetry definite integral

Let $$ {\rm I}(a,b) = \int_0^1 t^{a}(1-t)^{b} \, \d t \; \qquad (a\ge0,\ b\ge0) .$$

Show that ${\rm I}(a,b)={\rm I}(b,a)$,
Show that ${\rm I}(a,b)={\rm I}(a+1,b)+{\rm I}(a,b+1)$.
Show that $(a+1){\rm I}(a,b)=b{\rm I}(a+1,b-1)$ when $a$ and $b$ are positive and hence calculate ${\rm I}(a,b)$ when $a$ and $b$ are positive integers.

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1998 Paper 3 Q3

D: 1700.0 B: 1484.0

sequences recurrence relation first order linear recurrence geometric series limiting case financial mathematics particular solution

The value $V_N$ of a bond after $N$ days is determined by the equation $$ V_{N+1} = (1+c) V_{N} -d \qquad (c>0, \ d>0), $$ where $c$ and $d$ are given constants. By looking for solutions of the form $V_T= A k^T + B$ for some constants $A,B$ and $k$, or otherwise, find $V_N$ in terms of $V_0$. What is the solution for $c=0$? Show that this is the limit (for fixed $N$) as $c\rightarrow 0$ of your solution for $c>0$.

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1997 Paper 1 Q3

D: 1484.0 B: 1501.4

sequences recurrence relation exponentials and logarithms powers of 3 modular arithmetic estimation decimal approximation

Let $a_{1}=3$, $a_{n+1}=a_{n}^{3}$ for $n\geqslant 1$. (Thus $a_{2}=3^{3}$, $a_{3}=(3^{3})^{3}$ and so on.)

What digit appears in the unit place of $a_{7}$?
Show that $a_{7}\geqslant 10^{100}$.
What is $\dfrac{a_{7}+1}{2a_{7}}$ correct to two places of decimals? Justify your answer.

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1997 Paper 2 Q2

D: 1600.0 B: 1464.0

proof by induction sequences recurrence relation algebraic manipulation pattern recognition closed form expression

Suppose that $$3=\frac{2}{ x_1}=x_1+\frac{2}{ x_2} =x_2+\frac{2}{ x_3}=x_3+\frac{2}{ x_4}=\cdots.$$ Guess an expression, in terms of $n$, for $x_n$. Then, by induction or otherwise, prove the correctness of your guess.

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