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1995 Paper 1 Q3
D: 1500.0 B: 1500.0
sequences telescoping sum summation cube numbers method of differences alternating series polynomial identity

  1. If \(\mathrm{f}(r)\) is a function defined for \(r=0,1,2,3,\ldots,\) show that \[ \sum_{r=1}^{n}\left\{ \mathrm{f}(r)-\mathrm{f}(r-1)\right\} =\mathrm{f}(n)-\mathrm{f}(0). \]
  2. If \(\mathrm{f}(r)=r^{2}(r+1)^{2},\) evaluate \(\mathrm{f}(r)-\mathrm{f}(r-1)\) and hence determine \({\displaystyle \sum_{r=1}^{n}r^{3}.}\)
  3. Find the sum of the series \(1^{3}-2^{3}+3^{3}-4^{3}+\cdots+(2n+1)^{3}.\)

Solution:

  1. \(\,\) \begin{align*} && \sum_{r=1}^n \left (f(r) - f(r-1) \right) &= \cancel{f(1)} - f(0) + \cdots\\ &&&\quad\, \cancel{\f(2)}-\cancel{f(1)} + \cdots \\ &&&\quad\, \cancel{f(3)}-\cancel{f(2)} + \cdots \\ &&&\quad\, +\cdots + \cdots \\ &&&\quad\, \cancel{f(n-1)}-\cancel{f(n-2)} + \cdots \\ &&&\quad\, f(n)-\cancel{f(n-1)} \\ &&&=f(n) - f(0) \end{align*}
  2. If \(f(r) = r^2(r+1)^2\) then \begin{align*} && f(r) - f(r-1) &= r^2(r+1)^2 - (r-1)r^2 \\ &&&= r^2((r+1)^2-(r-1)^2) \\ &&&=4r^3 \end{align*} Therefore \begin{align*} && \sum_{r=1}^n 4r^3 &= n^2(n+1)^2-0 \\ \Rightarrow && \sum_{r=1}^n r^3 &= \frac{n^2(n+1)^2}{4} \\ \end{align*}
  3. So \begin{align*} && \sum_{r=1}^{2n+1} (-1)^{r-1}r^3 &= \sum_{r=1}^{2n+1} r^3 - 2 \sum_{r=1}^n (2r)^3 \\ &&&= \frac14(2n+1)^2(2n+2)^2 - 8 \frac{n^2(n+1)^2}{4} \\ &&&= (2n+1)^2(n+1)^2 - 2n^2(n+1)^2 \\ &&&= (n+1)^2(4n^2+4n+1 - 2n^2) \\ &&&= (n+1)^2(2n^2+4n+1) \end{align*}

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1995 Paper 1 Q13
D: 1500.0 B: 1484.0
Poisson distribution thinning property probability generating function conditional probability compound distribution order statistics summation

A scientist is checking a sequence of microscope slides for cancerous cells, marking each cancerous cell that she detects with a red dye. The number of cancerous cells on a slide is random and has a Poisson distribution with mean \(\mu.\) The probability that the scientist spots any one cancerous cell is \(p\), and is independent of the probability that she spots any other one.

  1. Show that the number of cancerous cells which she marks on a single slide has a Poisson distribution of mean \(p\mu.\)
  2. Show that the probability \(Q\) that the second cancerous cell which she marks is on the \(k\)th slide is given by \[ Q=\mathrm{e}^{-\mu p(k-1)}\left\{ (1+k\mu p)(1-\mathrm{e}^{-\mu p})-\mu p\right\} . \]

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1995 Paper 2 Q1
D: 1600.0 B: 1484.0
geometric series differentiation alternating series polynomial identity summation substitution x=-1 second derivative

  1. By considering \((1+x+x^{2}+\cdots+x^{n})(1-x)\) show that, if \(x\neq1\), \[ 1+x+x^{2}+\cdots+x^{n}=\frac{1-x^{n+1}}{1-x}. \]
  2. By differentiating both sides and setting \(x=-1\) show that \[ 1-2+3-4+\cdots+(-1)^{n-1}n \] takes the value \(-n/2\) is \(n\) is even and the value \((n+1)/2\) if \(n\) is odd.
  3. Show that \[ 1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n-1}n^{2}=(-1)^{n-1}(An^{2}+Bn) \] where the constants \(A\) and \(B\) are to be determined.

Solution:

  1. \begin{align*} && (1+x+x^{2}+\cdots+x^{n})(1-x) &= 1-x+x-x^2+\cdots -x^n+x^n-x^{n+1} \\ &&&= 1-x^{n+1} \\ \Rightarrow && 1+x+x^2+\cdots+x^n &= \frac{1-x^{n+1}}{1-x} \tag{dividing by \(1-x\)} \end{align*}
  2. \begin{align*} \frac{\d}{\d x}: && 0+1+2x+\cdots+nx^{n-1} &= \frac{(n+1)(1-x)x^n+(1-x^{n+1})}{(1-x)^2} \\ \Rightarrow && 1-2x+\cdots+(-1)^n n &= \frac{-(n+1)2(-1)^n+(1-(-1)^{n+1})}{4} \\ &&&= \begin{cases} \frac{-(n+1)\cdot2\cdot1+(1-(-1)}{4} & \text{if }n\text{ is even} \\ \frac{-(n+1)\cdot 2 \cdot(-1)+(1-1)}{4} & \text{if }n\text{ is odd}\end{cases} \\ &&&= \begin{cases} \frac{-n}{2} & \text{if }n\text{ is even}\\ \frac{n+1}{2} & \text{if }n\text{ is odd}\end{cases} \\ \end{align*}
  3. \begin{align*} x: && x+2x^2+\cdots+nx^{n} &= \frac{(n+1)(1-x)x^{n+1}+x(1-x^{n+1})}{(1-x)^2} \\ &&&= \frac{x+(n+1)x^{n+1}-nx^{n+2}}{(1-x)^2}\\ \frac{\d}{\d x}: && 1^2+2^2x + \cdots + n^2x^{n-1} &= \frac{(1-x)^2(1+(n+1)^2x^{n}-n(n+2)x^{n+1}) +2(1-x)(x+(n+1)x^{n+1}-nx^{n+2})}{(1-x)^4} \\ &&&= \frac{1 + x - (1 + n)^2 x^n + (2 n^2+2n-1) x^{n+1} - n^2 x^{n+2}}{(1-x)^3} \\ \Rightarrow && 1^2-2^2 + \cdots + (-1)^{n-1}n^2 &= \frac{(-1)^n \l - (1 + n)^2- (2 n^2+2n-1) - n^2 \r}{8} \\ &&&= \frac{(-1)^n(-4n^2-4n}{8} \\ &&&= \frac{(-1)^{n-1}(n^2+n)}{2} \end{align*}

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1995 Paper 3 Q4
D: 1700.0 B: 1499.3
trigonometry summation geometric series complex numbers inequality De Moivre trigonometric sums

Let \[ \mathrm{C}_{n}(\theta)=\sum_{k=0}^{n}\cos k\theta \] and let \[ \mathrm{S}_{n}(\theta)=\sum_{k=0}^{n}\sin k\theta, \] where \(n\) is a positive integer and \(0<\theta<2\pi.\) Show that \[ \mathrm{C}_{n}(\theta)=\frac{\cos(\tfrac{1}{2}n\theta)\sin\left(\frac{1}{2}(n+1)\theta\right)}{\sin(\frac{1}{2}\theta)}, \] and obtain the corresponding expression for \(\mathrm{S}_{n}(\theta)\). Hence, or otherwise, show that for \(0<\theta<2\pi,\) \[ \left|\mathrm{C}_{n}(\theta)-\frac{1}{2}\right|\leqslant\frac{1}{2\sin(\frac{1}{2}\theta)}. \]

Solution: \begin{align*} && C_n(\theta) &= \sum_{k=0}^n \cos k \theta \\ &&&= \textrm{Re} \left ( \sum_{k=0}^n \exp (ik \theta)\right)\\ &&&= \textrm{Re} \left ( \frac{e^{i(n+1)\theta}-1}{e^{i\theta}-1}\right)\\ &&&= \textrm{Re} \left ( \frac{e^{i(n+1)\theta/2}}{e^{i\theta/2}}\frac{e^{i(n+1)\theta/2}-e^{-i(n+1)\theta/2}}{e^{i\theta/2}-e^{-i\theta/2}}\right)\\ &&&= \textrm{Re} \left ( e^{in\theta/2}\frac{\sin \left ( (n+1)\theta/2 \right)}{\sin \left ( \theta/2 \right)}\right)\\ &&&= \frac{\sin \left ( (n+1)\theta/2 \right)}{\sin \left ( \theta/2 \right)}\textrm{Re} \left ( e^{in\theta/2}\right)\\ &&&= \frac{\sin \left ( (n+1)\theta/2 \right)}{\sin \left ( \theta/2 \right)}\cos \left ( \frac12n\theta\right)\\ \\ && S_n(\theta) &= \sum_{k=0}^n \sin k \theta \\ &&&= \textrm{Im} \left ( \sum_{k=0}^n \exp (ik \theta)\right)\\ &&&= \frac{\sin \left ( (n+1)\theta/2 \right)}{\sin \left ( \theta/2 \right)}\textrm{Im} \left ( e^{in\theta/2}\right)\\ &&&= \frac{\sin \left ( (n+1)\theta/2 \right)}{\sin \left ( \theta/2 \right)}\sin\left ( \frac12n\theta\right)\\ \\ && C_n(\theta) - \frac12 &= \frac{\sin \left ( (n+1)\theta/2 \right)}{\sin \left ( \theta/2 \right)}\cos \left ( \frac12n\theta\right) - \frac12 \\ &&&= \frac{2\sin \left ( (n+1)\theta/2 \right)\cos\left ( n\theta/2 \right)-\sin (\theta/2)}{2 \sin (\theta/2)} \\ &&&= \frac{\sin\left ( (n+1)\theta/2+n\theta/2\right)+\sin\left ( (n+1)\theta/2-n\theta/2\right)-\sin (\theta/2)}{2 \sin (\theta/2)} \\ &&&= \frac{\sin\left ( (n+1)\theta/2+n\theta/2\right)+\sin\left ( \theta/2\right)-\sin (\theta/2)}{2 \sin (\theta/2)} \\ &&&= \frac{\sin\left ( (2n+1)\theta/2\right)}{2 \sin (\theta/2)} \leqslant\frac{1}{2 \sin (\theta/2)} \\ \end{align*}

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1994 Paper 1 Q7
D: 1500.0 B: 1502.1
sequences proof by induction summation arithmetic series cubes pattern recognition telescoping

From the facts \begin{alignat*}{2} 1 & \quad=\quad & & 0\\ 2+3+4 & \quad=\quad & & 1+8\\ 5+6+7+8+9 & \quad=\quad & & 8+27\\ 10+11+12+13+14+15+16 & \quad=\quad & & 27+64 \end{alignat*} guess a general law. Prove it. Hence, or otherwise, prove that \[ 1^{3}+2^{3}+3^{3}+\cdots+N^{3}=\tfrac{1}{4}N^{2}(N+1)^{2} \] for every positive integer \(N\). [Hint. You may assume that \(1+2+3+\cdots+n=\frac{1}{2}n(n+1)\).]

Solution: \begin{align*} && (n^2+1) + (n^2+2) + \cdots + (n+1)^2 &= n^3+(n+1)^3 \\ \Leftrightarrow && \sum_{i=n^2+1}^{(n+1)^2} i &= n^3 + (n+1)^3 \\ && \sum_{i=n^2+1}^{(n+1)^2} i &= \sum_{i=1}^{(n+1)^2} i- \sum_{i=1}^{n^2} i \\ &&&= \frac{(n+1)^2((n+1)^2+1)}{2} - \frac{n^2(n^2+1)}{2} \\ &&&= \frac{(n+1)^2(n^2+2n+2) - n^2(n^2+1)}{2} \\ &&&= \frac{2(n+1)^3+n^2(n^2+2n+1) - n^2(n^2+1)}{2}\\ &&&= \frac{2(n+1)^3+2n^3 + n^2(n^2+1) - n^2(n^2+1)}{2}\\ &&&= (n+1)^3+n^3 \end{align*} \begin{align*} && \sum_{i=1}^{N^2} i &=(0^3+1^3)+ (1^3+2^3)+(2^3+3^3) + \cdots + ((N-1)^3+N^3) \\ &&&= 2 \left (1^3+2^3 + 3^3 + \cdots + (N-1)^3 \right) + N^3 \\ \Rightarrow && \sum_{i=1}^N i^3 &= \frac12 \left ( N^3+ \sum_{i=1}^{N^2} i \right) \\ &&&= \frac12 \left ( N^3 + \frac{N^2(N^2+1)}{2} \right) \\ &&&= \frac{N^2(N^2+1)+2N^3}{4} \\ &&&= \frac{N^2(N^2+2N+1)}{4} \\ &&&= \frac{N^2(N+1)^2}{4} \\ \end{align*}

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1993 Paper 2 Q15
D: 1600.0 B: 1501.4
probability continuous uniform distribution central limit theorem normal approximation rounding error summation variance calculation

Two computers, LEP and VOZ are programmed to add numbers after first approximating each number by an integer. LEP approximates the numbers by rounding: that is, it replaces each number by the nearest integer. VOZ approximates by truncation: that is, it replaces each number by the largest integer less than or equal to the number. The fractional parts of the numbers to be added are uniformly and independently distributed. (The fractional part of a number \(a\) is \(a-\left\lfloor a\right\rfloor ,\) where \(\left\lfloor a\right\rfloor \) is the largest integer less than or equal to \(a\).) Both computers approximate and add 1500 numbers. For each computer, find the probability that the magnitude of error in the answer will exceed 15. How many additions can LEP perform before the probability that the magnitude of error is less than 10 drops below 0.9?

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1993 Paper 3 Q4
D: 1700.0 B: 1500.0
sequences infinite series summation power series partial fractions geometric series logarithmic series telescoping series differentiation of series

Sum the following infinite series.

  1. \[ 1 + \frac13 \bigg({\frac12}\bigg)^2 +\frac15\bigg(\frac12\bigg)^4 + \cdots + \frac{1}{2n+1} \bigg(\frac12\bigg)^{2n} + \cdots \] .
  2. \[ 2 -x -x^3 +2x^4 - \cdots + 2x^{4k} - x^{4k+1} - x^{4k+3} +\cdots \] where \(|x| < 1\).
  3. \[ \sum _{r=2}^\infty {r\, 2^{r-2} \over 3^{r-1} } \].
  4. \[ \sum_{r=2}^\infty {2 \over r(r^2-1) } \].

Solution:

  1. \begin{align*} && \sum_{i=0}^{\infty} x^{2i+1}&= \frac{x}{1-x^2} \\ \Rightarrow &&&=\frac12 \left ( \frac{1}{1-x} - \frac{1}{1+x} \right) \\ \underbrace{\Rightarrow}_{\int} && \sum_{i=0}^{\infty} \frac{1}{2i+1} x^{2i+2} &= \frac12 \left ( -\ln (1-x) - \ln(1+x) \right) \\ \underbrace{\Rightarrow}_{x = 1/2} && \sum_{i=0}^\infty \frac{1}{2i+1} \left (\frac12 \right)^{2i+2} &= -\frac12 \ln \frac12 - \frac12 \ln \frac32 \\ &&&= -\frac12 \ln \frac34 \\ &&\frac14\sum_{i=0}^\infty \frac{1}{2i+1} \left (\frac12 \right)^{2i}&= \frac12 \ln \frac43 \\ \Rightarrow&& S &= 2 \ln \frac43 \end{align*}
  2. \begin{align*} \sum_{k=0}^{\infty} \left (2x^{4k} - x^{4k+1} - x^{4k+3} \right) &= \sum_{k=0}^{\infty} \left (2 - x^{1} - x^{3} \right) x^{4k} \\ &= \frac{2-x-x^3}{1-x^4} \\ &= \frac{(1-x)(2+x+x^2)}{(1-x)(1+x+x^2+x^3)} \\ &= \frac{2+x+x^2}{1+x+x^2+x^3} \end{align*}
  3. \begin{align*} && \frac{1}{(1-x)^2} &= \sum_{r=0}^{\infty} r x^{r-1} \\ \Rightarrow && 9 &= \sum_{r=1}^{\infty} r \left ( \frac23 \right)^{r-1} \\ \Rightarrow && \sum_{r=2}^{\infty} r \left ( \frac{2^{r-2}}{3^{r-1}} \right) &= \frac12 \left ( 9 - 1 \right) \\ &&&= 4 \end{align*}
  4. \begin{align*} && \frac{2}{r(r^2-1)} &= \frac{1}{r-1} - \frac{2}{r} + \frac{1}{r+1} \\ \Rightarrow && \sum_{r=2}^{\infty} \left ( \frac{1}{r-1} - \frac{2}{r} + \frac{1}{r+1} \right) &= \sum_{r=2}^{\infty} \left ( \frac{1}{r-1} - \frac{1}{r} - \frac{1}{r} + \frac{1}{r+1} \right) \\ &&&= \sum_{r=2}^{\infty} \left ( \frac{1}{r-1} - \frac{1}{r} \right)-\sum_{r=2}^{\infty} \left ( \frac{1}{r} - \frac{1}{r+1} \right) \\ &&&= 1 - \frac12 \\ &&&= \frac12 \end{align*}

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1993 Paper 3 Q15
D: 1700.0 B: 1501.5
probability negative binomial distribution expectation coin tossing geometric series summation waiting time

The probability of throwing a head with a certain coin is \(p\) and the probability of throwing a tail is \(q=1-p\). The coin is thrown until at least two heads and at least two tails have been thrown; this happens when the coin has been thrown \(N\) times. Write down an expression for the probability that \(N=n\). Show that the expectation of \(N\) is $$ 2\bigg({1\over pq} -1-pq\bigg). $$

Solution: This can either occur via \(N-2\) heads and \(1\) tail in the first \(N-1\) flips, followed by a tail, or \(N-2\) tails and \(1\) head in the first \(N-1\) flips, followed by another head, ie \begin{align*} \mathbb{P}(N = n) &= \underbrace{\binom{n-1}{1}}_{\text{ways to choose when the first tail occurs}}p^{n-2}q^2 + \underbrace{\binom{n-1}{1}}_{\text{ways to choose when the first head occurs}}q^{n-2}p^2 \\ &= (n-1)p^2q^2(p^{n-4}+q^{n-4}) \\ \\ \mathbb{E}(N) &= \sum_{n=4}^{\infty} n \cdot \mathbb{P}(N = n) \\ &= \sum_{n=4}^{\infty} n \cdot (n-1)p^2q^2(p^{n-4}+q^{n-4}) \\ &= \sum_{n=4}^{\infty} n \cdot (n-1)(p^{n-2}q^2+q^{n-2}p^2) \\ &= q^2\sum_{n=4}^{\infty} n(n-1)p^{n-2}+p^2\sum_{n=4}^{\infty} n(n-1)q^{n-2} \\ &= q^2\left ( \sum_{n=2}^{\infty} n(n-1)p^{n-2} -2 \cdot 1 - 3 \cdot 2 \cdot p\right)+p^2\left ( \sum_{n=2}^{\infty} n(n-1)q^{n-2} - 2-6q\right) \\ &= q^2\left ( 2(1-p)^{-3} -2 - 6 p\right)+p^2\left ( 2(1-q)^{-3} - 2-6q\right) \\ &= q^2\left ( 2q^{-3} -2 - 6 p\right)+p^2\left ( 2p^{-3} - 2-6q\right) \\ &= \frac{2}{q} - 2q^2 - 6pq^2+\frac{2}{p} -2p^2-6p^2q \\ &= \frac{2}{q}+\frac2p - 2(p^2+q^2) - 6pq \\ &= \frac{2}{pq} - 2((p+q)^2-2pq) - 6pq \\ &= \frac{2}{pq} - 2 -2pq \\ &= 2 \left (\frac1{pq} - 1 - pq \right) \end{align*}

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1992 Paper 3 Q8
D: 1700.0 B: 1515.1
complex numbers De Moivre trig identity roots of polynomials binomial coefficients summation Vieta's formulas cot cosec

Show that \[ \sin(2n+1)\theta=\sin^{2n+1}\theta\sum_{r=0}^{n}(-1)^{n-r}\binom{2n+1}{2r}\cot^{2r}\theta, \] where \(n\) is a positive integer. Deduce that the equation \[ \sum_{r=0}^{n}(-1)^{r}\binom{2n+1}{2r}x^{r}=0 \] has roots \(\cot^{2}(k\pi/(2n+1))\) for \(k=1,2,\ldots,n\). Show that

  • sep}{3mm}
  • \(\bf (i)\) \({\displaystyle \sum_{k=1}^{n}\cot^{2}\left(\frac{k\pi}{2n+1}\right)=\frac{n(2n-1)}{3}},\)
  • \(\bf (ii)\) \({\displaystyle \sum_{k=1}^{n}\tan^{2}\left(\frac{k\pi}{2n+1}\right)=n(2n+1)},\)
  • \(\bf (iii)\) \({\displaystyle \sum_{k=1}^{n}\mathrm{cosec}^{2}\left(\frac{k\pi}{2n+1}\right)=\frac{2n(n+1)}{3}}.\)

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1991 Paper 1 Q3
D: 1516.0 B: 1500.0
complex numbers Argand diagram geometric series rotation scaling geometric progression summation imaginary part

A path is made up in the Argand diagram of a series of straight line segments \(P_{1}P_{2},\) \(P_{2}P_{3},\) \(P_{3}P_{4},\ldots\) such that each segment is \(d\) times as long as the previous one, \((d\neq1)\), and the angle between one segment and the next is always \(\theta\) (where the segments are directed from \(P_{j}\) towards \(P_{j+1}\), and all angles are measured in the anticlockwise direction). If \(P_{j}\) represents the complex number \(z_{j},\) express \[ \frac{z_{n+1}-z_{n}}{z_{n}-z_{n-1}} \] as a complex number (for each \(n\geqslant2\)), briefly justifying your answer. If \(z_{1}=0\) and \(z_{2}=1\), obtain an expression for \(z_{n+1}\) when \(n\geqslant2\). By considering its imaginary part, or otherwise, show that if \(\theta=\frac{1}{3}\pi\) and \(d=2\), then the path crosses the real axis infinitely often.

Solution: \begin{align*} && | \frac{z_{n+1}-z_{n}}{z_{n}-z_{n-1}} | &= d \\ && \arg \left ( \frac{z_{n+1}-z_{n}}{z_{n}-z_{n-1}} \right) &= \arg (z_{n+1}-z_{n}) - \arg(z_{n}-z_{n-1}) \\ &&&= \theta \\ \Rightarrow && \frac{z_{n+1}-z_{n}}{z_{n}-z_{n-1}} &= d e^{i \theta} \end{align*} \begin{align*} && z_1 &= 0 \\ && z_2 &= 1 \\ && \frac{z_3-z_2}{z_2-z_1} &= de^{i \theta} \\ \Rightarrow && z_3 &= de^{i \theta} + 1 \\ && \frac{z_4-z_3}{z_3-z_2} &= de^{i \theta} \\ \Rightarrow && z_4 &= (d e^{i \theta})^2 + d e^{i \theta} + 1\\ \Rightarrow && z_{n+1} &= \frac{(de^{i \theta})^{n}-1}{de^{i \theta}-1} \end{align*} If \(d = 2, \theta = \tfrac13 \pi\), then, \(2e^{i \tfrac13 \pi} = 1 + \sqrt{3}i\) \begin{align*} \textrm{Im}(z_{n+1})) &= \textrm{Im} \left ( \frac{(2e^{i \tfrac13 \pi})^{n}-1}{2e^{i \tfrac13 \pi}-1}\right) \\ &= \textrm{Im} \left ( \frac{(2e^{i \tfrac13 \pi})^{n}-1}{\sqrt{3}i}\right) \\ &= -\frac{1}{\sqrt{3}}\textrm{Re} \left (2^n e^{i \frac{n}{3} \pi} \right) + \frac{1}{\sqrt{3}} \end{align*} Which clearly changes sign infinitely many times, ie crosses the origin infinitely many times.

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1991 Paper 1 Q9
D: 1500.0 B: 1516.0
proof inequality exponentials harmonic series summation bounding arguments proof by contradiction

  1. Suppose that the real number \(x\) satisfies the \(n\) inequalities \begin{alignat*}{2} 1<\ & x & & < 2\\ 2<\ & x^{2} & & < 3\\ 3<\ & x^{3} & & < 4\\ & \vdots\\ n<\ & x^{n} & & < n+1 \end{alignat*} Prove without the use of a calculator that \(n\leqslant4\).
  2. If \(n\) is an integer strictly greater than 1, by considering how many terms there are in \[ \frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n^{2}}, \] or otherwise, show that \[ \frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{n^{2}}>1. \] Hence or otherwise find, with justification, an integer \(N\) such that \({\displaystyle {\displaystyle \sum_{n=1}^{N}\frac{1}{n}>10.}}\)

Solution:

  1. Suppose \(n > 4\) then the following inequalities are both true \begin{align*} 3 < x^3 < 4 & \Rightarrow 3^5 < x^{15} < 4^{5}\\ 5 < x^5 < 6 & \Rightarrow 5^{3} < x^{15} < 6^3 \end{align*} But \(3^5 = 243\) and \(6^3 = 216\) so \(243 < x^{15} < 216\) whichis a contradiction.
  2. This question is wrong. Consider \(n = 2\), then \(\frac{1}{2+1} + \frac{1}{2+2} = \frac13+\frac14 = \frac{7}{12} < 1\). The question should be about \(n \geq 4\). \begin{align*} \frac{1}{n+1}+\frac1{n+2}+\cdots + \frac{1}{2n} > \frac{n}{2n} &= \frac12 \\ \frac{1}{2n+1}+\frac1{2n+2}+\cdots + \frac{1}{3n} > \frac{n}{3n} &= \frac13 \\ \frac{1}{4n+1}+\frac1{4n+2}+\cdots + \frac{1}{4n} > \frac{n}{4n} &= \frac14 \\ \sum_{k=1}^{n^2-n} \frac{1}{n+k} > \frac{13}{12} &> 1 \end{align*} We have a stronger result, \(\frac1{n+1} + \cdots + \frac1{4n} > 1\) for \(n > 4\) so we can take \(N = 4^{10}\) since, since there will be \(9\) sequences from \(\frac{1}{4^{i}+1} \to \frac{1}{4^{i+1}}\) and we will have \(\frac1{1}\) at the start to give use the extra \(1\).

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1991 Paper 1 Q14
D: 1516.0 B: 1457.1
probability combinatorics triangle inequality counting summation arithmetic series enumeration

A set of \(2N+1\) rods consists of one of each length \(1,2,\ldots,2N,2N+1\), where \(N\) is an integer greater than 1. Three different rods are selected from the set. Suppose their lengths are \(a,b\) and \(c\), where \(a > b > c\). Given that \(a\) is even and fixed, show, by considering the possible values of \(b\), that the number of selections in which a triangle can then be formed from the three rods is \[ 1+3+5+\cdots+(a-3), \] where we allow only non-degenerate triangles (i.e. triangles with non-zero area). Similarly obtain the number of selections in which a triangle may be formed when \(a\) takes some fixed odd value. Write down a formula for the number of ways of forming a non-degenerate triangle and verify it for \(N=3\). Hence show that, if three rods are drawn at random without replacement, then the probability that they can form a non-degenerate triangle is \[ \frac{(N-1)(4N+1)}{2(4N^{2}-1)}. \]

Solution: Suppose we have \(a = 2k\), it is necessary (by the triangle inequality) that \(b + c > a\). So the smallest \(b\) can be is \(k+1\), and then \(c\) must be \(k\) (1 choice). Then \(b\) could be \(k+2\) and \(c\) can be \(k+1\), \(k\), \(k-1\) (3 choices). Suppose \(b = k+i\) then \(c\) can be \(k+i-1, \ldots, k-i+1\) which is \(2i-1\) choices. This works until \(b = 2k-1\) and there are \(2(k-1)-1 = 2k-3 = a-3\) choices. Therefore there are \(1 + 3 + 5 + \cdots + (a-3)\) total choices. If \(a = 2k+1\) then, \(b = k+1\) is not possible \(b = k+2\) we have \(a = k+1, k\) (2 choices) \(b = k+3\) we have \(a = k+2, k+1, k, k-1\) (4 choices) \(b = k + i\) we have \(a = k+i-1, \cdots, k-i+2\) (\(2i-2\) choices) This works until \(b = k+k\) with \(2k-2 = a-3\) choices. So \(2 + 4 + \cdots + (a-3)\) If \(a\) is even, we have \(\left ( \frac{a-2}{2} \right)^2\) If \(a\) is odd we have \(\frac{(a-3)(a-1)}{4}\) Therefore the total number is: \begin{align*} C &= \sum_{k=2}^N \left ( \frac{(2k-2)^2}{4} + \frac{(2k+1-3)(2k+1-1)}{4} \right) \\ &= \sum_{k=2}^N \left ( (k-1)^2 + (k-1)k\right) \\ &= \sum_{k=2}^N (2k^2-3k+1) \\ &= \sum_{k=1}^N (2k^2-3k+1) \\ &= \frac{N(N+1)(2N+1)}{3} - \frac{3N(N+1)}{2} + N \\ &= \frac{N((N+1)(4N+2-9)+6)}{6} \\ &= \frac{N(4N+1)(N-1)}{6} \\ \end{align*} When \(N = 3\) we have \(1, 2, \cdots, 7\) sticks, and so \(a = 4\), \(1\) option \(a = 5\), \(2\) options \(a = 6\) \(4\) options \(a = 7\) \(6\) options for a total of \(13\). \(\frac{3 \cdot 13 \cdot 2}{6} = 13\) so this is promising, There are \(\binom{2N+1}{3}\) ways to choose three sticks (in order) and of those our formula tells us how many are valid, therefore \begin{align*} && P &= \frac{ \frac{N(4N+1)(N-1)}{6} }{\frac{(2N+1)2N(2N-1)}{6}} \\ &&&= \frac{(4N+1)(N-1)}{2(4N^2-1)} \end{align*}

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1991 Paper 2 Q15
D: 1600.0 B: 1484.0
probability uniform distribution expectation absolute difference summation discrete random variable random walk linearity of expectation

Integers \(n_{1},n_{2},\ldots,n_{r}\) (possibly the same) are chosen independently at random from the integers \(1,2,3,\ldots,m\). Show that the probability that \(\left|n_{1}-n_{2}\right|=k\), where \(1\leqslant k\leqslant m-1\), is \(2(m-k)/m^{2}\) and show that the expectation of \(\left|n_{1}-n_{2}\right|\) is \((m^{2}-1)/(3m)\). Verify, for the case \(m=2\), the result that the expection of \(\left|n_{1}-n_{2}\right|+\left|n_{2}-n_{3}\right|\) is \(2(m^{2}-1)/(3m).\) Write down the expectation, for general \(m\), of \[ \left|n_{1}-n_{2}\right|+\left|n_{2}-n_{3}\right|+\cdots+\left|n_{r-1}-n_{r}\right|. \] Desks in an examination hall are placed a distance \(d\) apart in straight lines. Each invigilator looks after one line of \(m\) desks. When called by a candidate, the invigilator walks to that candidate's desk, and stays there until called again. He or she is equally likely to be called by any of the \(m\) candidates in the line but candidates never call simultaneously or while the invigilator is attending to another call. At the beginning of the examination the invigilator stands by the first desk. Show that the expected distance walked by the invigilator in dealing with \(N+1\) calls is \[ \frac{d(m-1)}{6m}[2N(m+1)+3m]. \]

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1991 Paper 3 Q8
D: 1700.0 B: 1500.1
reduction formula integration by parts trigonometric binomial coefficients cos nθ summation proof

  1. The integral \(I_{k}\) is defined by \[ I_{k}=\int_{0}^{\theta}\cos^{k}x\,\cos kx\,\mathrm{d}x. \] Prove that \(2kI_{k}=kI_{k-1}+\cos^{k}\theta\sin k\theta.\)
  2. Prove that \[ 1+m\cos2\theta+\binom{m}{2}\cos4\theta+\cdots+\binom{m}{r}\cos2r\theta+\cdots+\cos2m\theta=2^{m}\cos^{m}\theta\cos m\theta. \]
  3. Using the results of (i) and (ii), show that \[ m\frac{\sin2\theta}{2}+\binom{m}{2}\frac{\sin4\theta}{4}+\cdots+\binom{m}{r}\frac{\sin2r\theta}{2r}+\cdots+\frac{\sin2m\theta}{2m} \] is equal to \[ \cos\theta\sin\theta+\cos^{2}\theta\sin2\theta+\cdots+\frac{1}{r}2^{r-1}\cos^{r}\theta\sin r\theta+\cdots+\frac{1}{m}2^{m-1}\cos^{m}\theta\sin m\theta. \]

Solution:

  1. \begin{align*} kI_k &= \int_0^\theta \cos^k x k\cos kx \d x \\ &= \left [\cos^k x \sin kx \right]_0^\theta - \int_0^\theta k \cos^{k-1} x \sin x \sin k x \d x \\ &= \cos^k \theta \sin k \theta - k\int_0^\theta \cos^{k-1} x \sin x \sin k x \d x \\ &= \cos^k \theta \sin k \theta + k\int_0^\theta \cos^{k-1} x (\cos (k-1) x - \cos k x \cos x) \d x \\ &= \cos^k\theta \sin k \theta + k I_{k-1} - kI_k \\ \end{align*} So \(2kI_k = kI_{k-1} + \cos^k \sin k \theta\)
  2. \begin{align*} && \sum_{r=0}^m \binom{m}{r} \cos 2r \theta &= \textrm{Re} \left ( \sum_{r=0}^m \binom{m}{r} \exp(i 2r \theta)\right) \\ &&&= \textrm{Re} \left ( \sum_{r=0}^m \binom{m}{r} \exp(i 2 \theta)^r\right) \\ &&&= \textrm{Re} \left ( \left (1+e^{2i \theta} \right)^m\right) \\ &&&= \textrm{Re} \left (\left (e^{i\theta}(e^{i \theta} +e^{-i\theta})\right)^m\right) \\ &&&= \textrm{Re} \left (\left (e^{i\theta}2\cos \theta\right)^m\right) \\ &&&= \textrm{Re} \left (2^m \cos^m \theta e^{im\theta}\right) \\ &&&= 2^m \cos^m \theta \cos m\theta \\ \end{align*}
  3. \begin{align*} \sum_{r=1}^m \binom{m}{r}\frac{\sin 2r \theta}{2r} &= \int_0^\theta \sum_{r=1}^m \binom{m}{r}\cos 2r x\d x \\ &= \int_0^\theta \left (2^m \cos^m x\cos mx- 1 \right ) \d x\\ &= 2^m I_m - \theta \\ &= \frac{2^{m-1}}{m}\cos^m \theta \sin m \theta + 2^{m-1} I_{m-1} - \theta \\ &= \frac{2^{m-1}}{m}\cos^m \theta \sin m \theta + \frac{2^{m-2}}{m-1}\cos^m \theta \sin m \theta+2^{m-2} I_{m-2} - \theta \\ &= \sum_{r=0}^{m} \frac{2^{m-1-r}}{m-r} \cos^{m-r} \theta \sin (m-r) \theta + I_0 - \theta \\ &= \sum_{r=0}^{m} \frac{2^{m-1-r}}{m-r} \cos^{m-r} \theta \sin (m-r) \theta + \int_0^\theta 1 \cdot 1 \d \theta - \theta \\ &= \sum_{r=0}^{m} \frac{2^{m-1-r}}{m-r} \cos^{m-r} \theta \sin (m-r) \theta \\ &= \sum_{r=0}^{m} \frac{2^{r-1}}{r} \cos^{r} \theta \sin r \theta \end{align*}

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1991 Paper 3 Q16
D: 1700.0 B: 1504.3
probability bivariate data joint distribution marginal probability expectation summation discrete random variables integer constraints

The random variables \(X\) and \(Y\) take integer values \(x\) and \(y\) respectively which are restricted by \(x\geqslant1,\) \(y\geqslant1\) and \(2x+y\leqslant2a\) where \(a\) is an integer greater than 1. The joint probability is given by \[ \mathrm{P}(X=x,Y=y)=c(2x+y), \] where \(c\) is a positive constant, within this region and zero elsewhere. Obtain, in terms of \(x,c\) and \(a,\) the marginal probability \(\mathrm{P}(X=x)\) and show that \[ c=\frac{6}{a(a-1)(8a+5)}. \] Show that when \(y\) is an even number the marginal probability \(\mathrm{P}(Y=y)\) is \[ \frac{3(2a-y)(2a+2+y)}{2a(a-1)(8a+5)} \] and find the corresponding expression when \(y\) is off. Evaluate \(\mathrm{E}(Y)\) in terms of \(a\).

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