(a) Use standard results from the list of formulae (MF19) to show that

\(\sum_{r=1}^{n} (2-3r)(5-3r) = an^3 + bn^2 + cn,\)

where \(a, b\) and \(c\) are integers to be determined.

(b) Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(2-3r)(5-3r)}\) in terms of \(n\).

(c) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(2-3r)(5-3r)}\).

(a) Use standard results from the list of formulae (MF19) to show that

\(\sum_{r=1}^{n} (2-3r)(5-3r) = an^3 + bn^2 + cn,\)

where \(a, b\) and \(c\) are integers to be determined.

(b) Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(2-3r)(5-3r)}\) in terms of \(n\).

(c) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(2-3r)(5-3r)}\).

The quartic equation \(x^4 + 7x^2 + 3x + 22 = 0\) has roots \(\alpha, \beta, \gamma, \delta\).

(a) Find the value of \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\).

(b) Find the value of \(\alpha^4 + \beta^4 + \gamma^4 + \delta^4\).

(c) Use standard results from the list of formulae (MF19) to find the value of \(\sum_{r=1}^{10} ((\alpha^2 + r)^2 + (\beta^2 + r)^2 + (\gamma^2 + r)^2 + (\delta^2 + r)^2)\).

1. Use the List of formulae (MF19) to find \(\sum_{r=1}^{n} (2r+1)\) in terms of \(n\), simplifying your answer.
2. Show that \(\frac{2r+1}{(r^2+1)(r^2+2r+2)} = \frac{1}{r^2+1} - \frac{1}{r^2+2r+2}\).
3. Use the method of differences to find \(\sum_{r=1}^{n} \frac{2r+1}{(r^2+1)(r^2+2r+2)}\).
4. Deduce the value of \(\sum_{r=1}^{\infty} \frac{2r+1}{(r^2+1)(r^2+2r+2)}\).

(a) Use the method of differences to find \(\sum_{r=1}^{n} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant.

It is given that \(\sum_{r=1}^{\infty} \frac{5k}{(5r+k)(5r+5+k)} = \frac{1}{3}\).

(b) Find the value of \(k\).

(c) Hence find \(\sum_{r=n}^{n^2} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\).

(a) Use the method of differences to find \(\sum_{r=1}^{n} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant.

It is given that \(\sum_{r=1}^{\infty} \frac{5k}{(5r+k)(5r+5+k)} = \frac{1}{3}\).

(b) Find the value of \(k\).

(c) Hence find \(\sum_{r=n}^{n^2} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\).

(a) Use standard results from the list of formulae (MF19) to show that

\(\sum_{r=1}^{N} r(r+1)(3r+4) = \frac{1}{12}N(N+1)(N+2)(9N+19).\)

(b) Express \(\frac{3r+4}{r(r+1)}\) in partial fractions and hence use the method of differences to find

\(\sum_{r=1}^{N} \frac{3r+4}{r(r+1)} \left( \frac{1}{4} \right)^{r+1}\)

in terms of \(N\).

(c) Deduce the value of

\(\sum_{r=1}^{\infty} \frac{3r+4}{r(r+1)} \left( \frac{1}{4} \right)^{r+1}.\)

(a) Use standard results from the list of formulae (MF19) to show that

\(\sum_{r=1}^{N} r(r+1)(3r+4) = \frac{1}{12}N(N+1)(N+2)(9N+19).\)

(b) Express \(\frac{3r+4}{r(r+1)}\) in partial fractions and hence use the method of differences to find

\(\sum_{r=1}^{N} \frac{3r+4}{r(r+1)} \left( \frac{1}{4} \right)^{r+1}\)

in terms of \(N\).

(c) Deduce the value of

\(\sum_{r=1}^{\infty} \frac{3r+4}{r(r+1)} \left( \frac{1}{4} \right)^{r+1}.\)

(a) By considering \((r+1)^2 - r^2\), use the method of differences to prove that

\(\sum_{r=1}^{n} r = \frac{1}{2} n(n+1).\)

(b) Given that \(\sum_{r=1}^{n} (r+a) = n\), find \(a\) in terms of \(n\).

(a) Use standard results from the list of formulae (MF19) to find \(\sum_{r=1}^{n} (3r^2 + 3r + 1)\) in terms of \(n\), simplifying your answer.

(b) Show that \(\frac{1}{r^3} - \frac{1}{(r+1)^3} = \frac{3r^2 + 3r + 1}{r^3 (r+1)^3}\) and hence use the method of differences to find \(\sum_{r=1}^{n} \frac{3r^2 + 3r + 1}{r^3 (r+1)^3}\).

(c) Deduce the value of \(\sum_{r=1}^{\infty} \frac{3r^2 + 3r + 1}{r^3 (r+1)^3}\).

The cubic equation \(27x^3 + 18x^2 + 6x - 1 = 0\) has roots \(\alpha, \beta, \gamma\).

(a) Show that a cubic equation with roots \(3\alpha + 1, 3\beta + 1, 3\gamma + 1\) is \(y^3 - y^2 + y - 2 = 0\).

The sum \((3\alpha + 1)^n + (3\beta + 1)^n + (3\gamma + 1)^n\) is denoted by \(S_n\).

(b) Find the values of \(S_2\) and \(S_3\).

(c) Find the values of \(S_{-1}\) and \(S_{-2}\).

(a) By considering \((r+1)^2 - r^2\), use the method of differences to prove that

\(\sum_{r=1}^{n} r = \frac{1}{2} n(n+1).\)

(b) Given that \(\sum_{r=1}^{n} (r+a) = n\), find \(a\) in terms of \(n\).

The cubic equation \(x^3 + 4x^2 + 6x + 1 = 0\) has roots \(\alpha, \beta, \gamma\).

(a) Find the value of \(\alpha^2 + \beta^2 + \gamma^2\).

(b) Use standard results from the list of formulae (MF19) to show that

\(\sum_{r=1}^{n} ((\alpha + r)^2 + (\beta + r)^2 + (\gamma + r)^2) = n(n^2 + an + b),\)

where \(a\) and \(b\) are constants to be determined.

3 (a) Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(kr+1)(kr-k+1)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant.

(b) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(kr+1)(kr-k+1)}\).

(c) Find also \(\sum_{r=n}^{n^2} \frac{1}{(kr+1)(kr-k+1)}\) in terms of \(n\) and \(k\).

The cubic equation \(x^3 + 4x^2 + 6x + 1 = 0\) has roots \(\alpha, \beta, \gamma\).

(a) Find the value of \(\alpha^2 + \beta^2 + \gamma^2\).

(b) Use standard results from the list of formulae (MF19) to show that

\(\sum_{r=1}^{n} ((\alpha + r)^2 + (\beta + r)^2 + (\gamma + r)^2) = n(n^2 + an + b),\)

where \(a\) and \(b\) are constants to be determined.

3 (a) Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(kr+1)(kr-k+1)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant.

(b) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(kr+1)(kr-k+1)}\).

(c) Find also \(\sum_{r=n}^{n^2} \frac{1}{(kr+1)(kr-k+1)}\) in terms of \(n\) and \(k\).

2 (a) Use standard results from the list of formulae (MF19) to show that

\(\sum_{r=1}^{n} (6r^2 + 6r - 5) = an^3 + bn^2 + cn\),

where \(a, b\) and \(c\) are integers to be determined.

(b) Use the method of differences to find \(\sum_{r=1}^{n} \frac{6r^2 + 6r - 5}{r^2 + r}\) in terms of \(n\).

(c) Find also \(\sum_{r=n+1}^{2n} \frac{6r^2 + 6r - 5}{r^2 + r}\) in terms of \(n\).

Let \(a\) be a positive constant.

(a) Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(ar+1)(ar+a+1)}\) in terms of \(n\) and \(a\).

(b) Find the value of \(a\) for which \(\sum_{r=1}^{\infty} \frac{1}{(ar+1)(ar+a+1)} = \frac{1}{6}\).

(a) By considering \((2r+1)^3 - (2r-1)^3\), use the method of differences to prove that \(\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1)\).

Let \(S_n = 1^2 + 3 \times 2^2 + 3^2 + 3 \times 4^2 + 5^2 + 3 \times 6^2 + \ldots + \left(2 + (-1)^n\right)n^2\).

(b) Show that \(S_{2n} = \frac{1}{3}n(2n+1)(an+b)\), where \(a\) and \(b\) are integers to be determined.

(c) State the value of \(\lim_{n \to \infty} \frac{S_{2n}}{n^3}\).

(a) Use the list of formulae (MF19) to find \(\sum_{r=1}^{n} r(r+2)\) in terms of \(n\), simplifying your answer.

(b) Express \(\frac{1}{r(r+2)}\) in partial fractions and hence find \(\sum_{r=1}^{n} \frac{1}{r(r+2)}\) in terms of \(n\).

(c) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{r(r+2)}\).

(a) Use standard results from the List of formulae (MF19) to find \(\sum_{r=1}^{n} (1 - r - r^2)\) in terms of \(n\), simplifying your answer.

(b) Show that \(\frac{1 - r - r^2}{(r^2 + 2r + 2)(r^2 + 1)} = \frac{r + 1}{(r+1)^2 + 1} - \frac{r}{r^2 + 1}\) and hence use the method of differences to find \(\sum_{r=1}^{n} \frac{1 - r - r^2}{(r^2 + 2r + 2)(r^2 + 1)}\).

(a) Use standard results from the list of formulae (MF19) to find \(\sum_{r=1}^{n} r(r+1)(r+2)\) in terms of \(n\), fully factorising your answer.

(b) Express \(\frac{1}{r(r+1)(r+2)}\) in partial fractions and hence use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{r(r+1)(r+2)}\).

(c) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{r(r+1)(r+2)}\).

The cubic equation \(x^3 + 2x^2 + 3x + 3 = 0\) has roots \(\alpha, \beta, \gamma\).

(a) Find the value of \(\alpha^2 + \beta^2 + \gamma^2\).

(b) Show that \(\alpha^3 + \beta^3 + \gamma^3 = 1\).

(c) Use standard results from the list of formulae (MF19) to show that

\(\sum_{r=1}^{n} ((\alpha + r)^3 + (\beta + r)^3 + (\gamma + r)^3) = n + \frac{1}{4}n(n+1)(an^2 + bn + c),\)

where \(a, b, c\) are constants to be determined.

2 (a) Use standard results from the List of Formulae (MF19) to show that
\(\sum_{r=1}^{n}(7 r+1)(7 r+8)=a n^{3}+b n^{2}+c n\)
where \(a, b\) and \(c\) are constants to be determined.

(b) Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(7 r+1)(7 r+8)}\) in terms of \(n\).

(c) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(7 r+1)(7 r+8)}\).

4 (a) By first expressing \(\frac{1}{r^{2}-1}\) in partial fractions, show that
\(\sum_{r=2}^{n} \frac{1}{r^{2}-1}=\frac{3}{4}-\frac{a n+b}{2 n(n+1)},\)
where \(a\) and \(b\) are integers to be found.

(b) Deduce the value of \(\sum_{r=2}^{\infty} \frac{1}{r^{2}-1}\).

(c) Find the limit, as \(n \rightarrow \infty\), of \(\sum_{r=n+1}^{2 n} \frac{n}{r^{2}-1}\).

4 (i) Use the method of differences to show that \(\sum_{r=1}^{N} \frac{1}{(3 r+1)(3 r-2)}=\frac{1}{3}-\frac{1}{3(3 N+1)}\).

(ii) Find the limit, as \(N \rightarrow \infty\), of \(\sum_{r=N+1}^{N^{2}} \frac{N}{(3 r+1)(3 r-2)}\).

\(5 \quad\) Let \(S_{N}=\sum_{r=1}^{N}(5 r+1)(5 r+6)\) and \(T_{N}=\sum_{r=1}^{N} \frac{1}{(5 r+1)(5 r+6)}\).
(i) Use standard results from the List of Formulae (MF10) to show that
\(S_{N}=\frac{1}{3} N\left(25 N^{2}+90 N+83\right)\)

(ii) Use the method of differences to express \(T_{N}\) in terms of \(N\).

monisinatin
(iii) Find \(\lim _{N \rightarrow \infty}\left(N^{-3} S_{N} T_{N}\right)\).

Let \(S_{n}=\sum_{r=1}^{n}(-1)^{r-1} r^{2}\).
(i) Use the standard result for \(\sum_{r=1}^{n} r^{2}\) given in the List of Formulae (MF10) to show that
\(S_{2 n}=-n(2 n+1) .\)

(ii) State the value of \(\lim _{n \rightarrow \infty} \frac{S_{2 n}}{n^{2}}\) and find \(\lim _{n \rightarrow \infty} \frac{S_{2 n+1}}{n^{2}}\).

Answer only one of the following two alternatives.

EITHER

(i) Show that if \(z=e^{i\theta}\) and \(z\neq -1\), then \(\frac{z-1}{z+1}=i\tan\frac{\theta}{2}\).

(ii) Hence, or otherwise, show that if \(z\) is a cube root of unity, then

\(\frac{z^3-1}{z^3+1}+\frac{z^2-1}{z^2+1}+\frac{z-1}{z+1}=0.\)

(iii) Hence write down three roots of

\((z^3-1)(z^2+1)(z+1)+(z^2-1)(z^3+1)(z+1)+(z-1)(z^3+1)(z^2+1)=0\)

and find the other three roots. Give your answers in an exact form.

OR

It is given that \(\mathbf e\) is an eigenvector of \(\mathbf A\), with corresponding eigenvalue \(\lambda\).

(i) Write down another eigenvector of \(\mathbf A\) corresponding to \(\lambda\).

(ii) Write down an eigenvector and corresponding eigenvalue of \(\mathbf A^n\), where \(n\) is a positive integer.

Let \(\mathbf A=\begin{pmatrix}3&0&0\\2&7&0\\4&8&1\end{pmatrix}\).

(iii) Find a matrix \(\mathbf P\) and a diagonal matrix \(\mathbf D\) such that \(\mathbf A^n=\mathbf P\mathbf D\mathbf P^{-1}\).

(iv) Determine the set of values of the real constant \(k\) such that \(\sum_{n=1}^{\infty} k^n(\mathbf A^n-k\mathbf A^{n+1})=k\mathbf A\).

(i) Verify that
\(\frac{n(\mathrm{e}-1)+\mathrm{e}}{n(n+1) \mathrm{e}^{n+1}}=\frac{1}{n \mathrm{e}^{n}}-\frac{1}{(n+1) \mathrm{e}^{n+1}} .\)

Let \(S_{N}=\sum_{n=1}^{N} \frac{n(\mathrm{e}-1)+\mathrm{e}}{n(n+1) \mathrm{e}^{n+1}}\).
(ii) Express \(S_{N}\) in terms of \(N\) and e.

Let \(S=\lim _{N \rightarrow \infty} S_{N}\).
(iii) Find the least value of \(N\) such that \((N+1)\left(S-S_{N}\right)\lt10^{-3}\).

Let
\(S_{N}=\sum_{r=1}^{N}(3 r+1)(3 r+4) \quad \text { and } \quad T_{N}=\sum_{r=1}^{N} \frac{1}{(3 r+1)(3 r+4)} .\)
(i) Use standard results from the List of Formulae (MF10) to show that
\(S_{N}=N\left(3 N^{2}+12 N+13\right)\)

(ii) Use the method of differences to show that
\(T_{N}=\frac{1}{12}-\frac{1}{3(3 N+4)}\)
(iii) Deduce that \(\frac{S_{N}}{T_{N}}\) is an integer.

(iv) Find \(\lim _{N \rightarrow \infty} \frac{S_{N}}{N^{3} T_{N}}\).

It is given that \(\sum_{r=1}^{n} u_{r}=n^{2}(2 n+3)\), where \(n\) is a positive integer.
(i) Find \(\sum_{r=n+1}^{2 n} u_{r}\).

(ii) Find \(u_{r}\).

(i) Verify that \(\frac{2 r+1}{r(r+1)(r+2)}=\frac{1}{2}\left\{\frac{(2 r+1)(2 r+3)}{(r+1)(r+2)}-\frac{(2 r-1)(2 r+1)}{r(r+1)}\right\}\).

(ii) Hence show that \(\sum_{r=1}^{n} \frac{2 r+1}{r(r+1)(r+2)}=\frac{1}{2}\left\{\frac{(2 n+1)(2 n+3)}{(n+1)(n+2)}-\frac{3}{2}\right\}\).

(iii) Deduce the value of \(\sum_{r=1}^{\infty} \frac{2 r+1}{r(r+1)(r+2)}\).

Expand and simplify \((r+1)^{4}-r^{4}\).

Use the method of differences together with the standard results for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^{2}\) to show that
\(\sum_{r=1}^{n} r^{3}=\frac{1}{4} n^{2}(n+1)^{2} .\)

Use the List of Formulae (MF10) to show that \(\sum_{r=1}^{13}\left(3 r^{2}-5 r+1\right)\) and \(\sum_{r=0}^{9}\left(r^{3}-1\right)\) have the same numerical value.

Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(2 r)^{2}-1}\).

Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(2 r)^{2}-1}\).

Verify that \(\dfrac{1}{(3r+1)(3r+4)}=\dfrac{1}{3}\left(\dfrac{1}{3r+1}-\dfrac{1}{3r+4}\right)\).

Let \(S_N\) denote \(\sum_{r=1}^{N}\dfrac{1}{(3r+1)(3r+4)}\) and let \(S\) denote \(\sum_{r=1}^{\infty}\dfrac{1}{(3r+1)(3r+4)}\). Find the least value of \(N\) such that \(S-S_N\lt\dfrac{1}{10000}\).

Express \(\frac{4}{r(r+1)(r+2)}\) in partial fractions and hence find \(\sum_{r=1}^{n} \frac{4}{r(r+1)(r+2)}\).

Deduce the value of \(\sum_{r=1}^{\infty} \frac{4}{r(r+1)(r+2)}\).

Find \(\sum_{r=1}^{n}(4 r-3)(4 r+1)\), giving your answer in its simplest form.

Given that
\(u_{k}=\frac{1}{\sqrt{2 k-1}}-\frac{1}{\sqrt{2 k+1}},\)
express \(\sum_{k=13}^{n} u_{k}\) in terms of \(n\).

Deduce the value of \(\sum_{k=13}^{\infty} u_{k}\).

Given that
\(u_{k}=\frac{1}{\sqrt{2 k-1}}-\frac{1}{\sqrt{2 k+1}},\)
express \(\sum_{k=13}^{n} u_{k}\) in terms of \(n\).

Deduce the value of \(\sum_{k=13}^{\infty} u_{k}\).

Use the method of differences to show that \(\sum_{r=1}^{N} \frac{1}{(2 r+1)(2 r+3)}=\frac{1}{6}-\frac{1}{2(2 N+3)}\).

Deduce that \(\sum_{r=N+1}^{2 N} \frac{1}{(2 r+1)(2 r+3)}\lt \frac{1}{8 N}\).

Let \(\mathrm{f}(r)=r!(r-1)\). Simplify \(\mathrm{f}(r+1)-\mathrm{f}(r)\) and hence find \(\sum_{r=n+1}^{2 n} r!\left(r^{2}+1\right)\).

It is given that
\(S_{n}=\sum_{r=1}^{n} u_{r}=2 n^{2}+n\)

Write down the values of \(S_{1}, S_{2}, S_{3}, S_{4}\). Express \(u_{r}\) in terms of \(r\), justifying your answer.

Find
\(\sum_{r=n+1}^{2 n} u_{r} .\)

It is given that
\(S_{n}=\sum_{r=1}^{n} u_{r}=2 n^{2}+n\)

Write down the values of \(S_{1}, S_{2}, S_{3}, S_{4}\). Express \(u_{r}\) in terms of \(r\), justifying your answer.

Find
\(\sum_{r=n+1}^{2 n} u_{r} .\)

Express \(\frac{1}{r(r+1)(r-1)}\) in partial fractions.

Find
\(\sum_{r=2}^{n} \frac{1}{r(r+1)(r-1)}\)

State the value of
\(\sum_{r=2}^{\infty} \frac{1}{r(r+1)(r-1)} .\)

Use induction to prove that
\(\sum_{n=1}^{N} \frac{4 n+1}{n(n+1)(2 n-1)(2 n+1)}=1-\frac{1}{(N+1)(2 N+1)}\)

Show that
\(\sum_{n=N+1}^{2 N} \frac{4 n+1}{n(n+1)(2 n-1)(2 n+1)}\lt \frac{3}{8 N^{2}} .\)

Express \(\frac{1}{(2 r+1)(2 r+3)}\) in partial fractions and hence use the method of differences to
\(\sum_{r=1}^{n} \frac{1}{(2 r+1)(2 r+3)}\)

Deduce the value of
\(\sum_{r=1}^{\infty} \frac{1}{(2 r+1)(2 r+3)}\)

Expand and simplify \((r+1)^{4}-r^{4}\).

Use the method of differences together with the standard results for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^{2}\) to show that
\(\sum_{r=1}^{n} r^{3}=\frac{1}{4} n^{2}(n+1)^{2} .\)

Show that \(\sum_{r=n+1}^{2 n} r^{2}=\frac{1}{6} n(2 n+1)(7 n+1)\).

The sum \(S_{N}\) is defined by \(S_{N}=\sum_{n=1}^{N} n^{5}\). Using the identity
\(\left(n+\frac{1}{2}\right)^{6}-\left(n-\frac{1}{2}\right)^{6} \equiv 6 n^{5}+5 n^{3}+\frac{3}{8} n\)
find \(S_{N}\) in terms of \(N\). [You need not simplify your result.]

Hence find \(\lim _{N \rightarrow \infty} N^{-\lambda} S_{N}\), for each of the two cases
(i) \(\lambda=6\),
(ii) \(\lambda\gt 6\).

Find \(2^{2}+4^{2}+\ldots+(2 n)^{2}\).
Hence find \(1^{2}-2^{2}+3^{2}-4^{2}+\ldots-(2 n)^{2}\), simplifying your answer.

\(\mathbf{1}\) Verify that \(\frac{1}{n^{2}}-\frac{1}{(n+1)^{2}}=\frac{2 n+1}{n^{2}(n+1)^{2}}\).
Let \(S_{N}=\sum_{r=1}^{N} \frac{2 r+1}{r^{2}(r+1)^{2}}\). Express \(S_{N}\) in terms of \(N\).
Let \(S=\lim _{N \rightarrow \infty} S_{N}\). Find the least value of \(N\) such that \(S-S_{N}\lt 10^{-16}\).