Prove by mathematical induction that, for every integer \(n \geq 2\),

\(\frac{d^n}{dx^n}(x \ln x) = (-1)^n (n-2)! x^{1-n}.\)

Prove by mathematical induction that, for all positive integers n,

\(\frac{d^n}{dx^n}(\arctan x) = P_n(x)(1+x^2)^{-n},\)

where \(P_n(x)\) is a polynomial of degree \(n-1\).

The sequence \(u_1, u_2, u_3, \ldots\) is such that \(u_1 = 4\) and \(u_{n+1} = 3u_n - 2\) for \(n \geq 1\).

Prove by induction that \(u_n = 3^n + 1\) for all positive integers \(n\).

Prove by mathematical induction that, for all positive integers n,

\(\frac{d^n}{dx^n} \left( \arctan x \right) = P_n(x) (1 + x^2)^{-n},\)

where \(P_n(x)\) is a polynomial of degree \(n - 1\).

(a) Prove by mathematical induction that, for all positive integers \(n\),

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

The sum \(S_n\) is defined by \(S_n = \sum_{r=1}^{n} r^4\).

(b) Using the identity

\((2r+1)^5 - (2r-1)^5 \equiv 160r^4 + 80r^2 + 2,\)

show that \(S_n = \frac{1}{30}n(n+1)(2n+1)(3n^2 + 3n - 1).\)

(c) Find the value of \(\lim_{n \to \infty} \left( n^{-5}S_n \right).\)

Prove by mathematical induction that, for all positive integers \(n\),

\(1 + 2x + 3x^2 + \ldots + nx^{n-1} = \frac{1 - (n+1)x^n + nx^{n+1}}{(1-x)^2}.\)

Prove by mathematical induction that, for all positive integers n,

\(\frac{d^n}{dx^n} \left( x^2 e^x \right) = \left( x^2 + 2nx + n(n-1) \right) e^x.\)

Prove by mathematical induction that, for all positive integers \(n\),

\(1 + 2x + 3x^2 + \ldots + nx^{n-1} = \frac{1 - (n+1)x^n + nx^{n+1}}{(1-x)^2}.\)

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

(a) Prove by mathematical induction that, for all positive integers \(n\),

\(2\mathbf{A}^n = \begin{pmatrix} 2 \times 3^n & 0 \\ 3^n - 1 & 2 \end{pmatrix}.\)

(b) Find, in terms of \(n\), the inverse of \(\mathbf{A}^n\).

Let \(A = \begin{pmatrix} 3 & 0 \\ 1 & 1 \end{pmatrix}\).

(a) Prove by mathematical induction that, for all positive integers \(n\),

\(2A^n = \begin{pmatrix} 2 \times 3^n & 0 \\ 3^n - 1 & 2 \end{pmatrix}.\)

(b) Find, in terms of \(n\), the inverse of \(A^n\).

The sequence of positive numbers \(u_1, u_2, u_3, \ldots\) is such that \(u_1 > 4\) and, for \(n \geq 1\),

\(u_{n+1} = \frac{u_n^2 + u_n + 12}{2u_n}.\)

(a) By considering \(u_{n+1} - 4\), or otherwise, prove by mathematical induction that \(u_n > 4\) for all positive integers \(n\). [5]

(b) Show that \(u_{n+1} < u_n\) for \(n \geq 1\). [3]

Let \(\mathbf{A} = \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix}\), where \(a\) is a positive constant.

(a) State the type of the geometrical transformation in the \(x-y\) plane represented by \(\mathbf{A}\). [1]

(b) Prove by mathematical induction that, for all positive integers \(n\),

\(\mathbf{A}^n = \begin{pmatrix} 1 & na \\ 0 & 1 \end{pmatrix}.\) [5]

Let \(\mathbf{B} = \begin{pmatrix} b & b \\ a^{-1} & a^{-1} \end{pmatrix}\), where \(b\) is a positive constant.

(c) Find the equations of the invariant lines, through the origin, of the transformation in the \(x-y\) plane represented by \(\mathbf{A}^n \mathbf{B}\). [6]

The function \(f\) is such that \(f''(x)= f(x)\)

Prove by mathematical induction that, for every positive integer n,

\(\frac{d^{2n-1}}{dx^{2n-1}}(xf(x)) = xf'(x) + (2n-1)f(x).\)

(a) Prove by mathematical induction that, for all positive integers \(n\),

\(\sum_{r=1}^{n} (5r^4 + r^2) = \frac{1}{2} n^2 (n+1)^2 (2n+1).\)

(b) Use the result given in part (a) together with the List of formulae (MF19) to find \(\sum_{r=1}^{n} r^4\) in terms of \(n\), fully factorising your answer.

The sequence of real numbers \(a_1, a_2, a_3, \ldots\) is such that \(a_1 = 1\) and

\(a_{n+1} = \left( a_n + \frac{1}{a_n} \right)^3.\)

(a) Prove by mathematical induction that \(\ln a_n \geq 3^{n-1} \ln 2\) for all integers \(n \geq 2\).

[You may use the fact that \(\ln \left( x + \frac{1}{x} \right) > \ln x\) for \(x > 0\).]

(b) Show that \(\ln a_{n+1} - \ln a_n > 3^{n-1} \ln 4\) for \(n \geq 2\).

It is given that \(y = xe^{ax}\), where \(a\) is a constant.

Prove by mathematical induction that, for all positive integers \(n\),

\(\frac{d^n y}{dx^n} = \left( a^n x + na^{n-1} \right) e^{ax}.\)

5 Prove by mathematical induction that, for every positive integer \(n\),
\(\frac{\mathrm{d}^{2 n-1}}{\mathrm{~d} x^{2 n-1}}(x \sin x)=(-1)^{n-1}(x \cos x+(2 n-1) \sin x)\)

\(6 \quad\) Let \(\mathbf{A}=\left(\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right)\).
(a) The transformation in the \(x-y\) plane represented by \(\mathbf{A}^{-1}\) transforms a triangle of area \(30 \mathrm{~cm}^{2}\) into a triangle of area \(d \mathrm{~cm}^{2}\).

Find the value of \(d\).

(b) Prove by mathematical induction that, for all positive integers \(n\),
\(\mathbf{A}^{n}=\left(\begin{array}{cc}
2^{n} & 0 \\
2^{n}-1 & 1
\end{array}\right) .\)

(c) The line \(y=2 x\) is invariant under the transformation in the \(x-y\) plane represented by \(\mathbf{A}^{n} \mathbf{B}\), where \(\mathbf{B}=\left(\begin{array}{rr}1 & 0 \\ 33 & 0\end{array}\right)\).

Find the value of \(n\).

8 (i) Prove by mathematical induction that, for \(z eq 1\) and all positive integers \(n\),
\(1+z+z^{2}+\ldots+z^{n-1}=\frac{z^{n}-1}{z-1}\)

(ii) By letting \(z=\frac{1}{2}(\cos \theta+\mathrm{i} \sin \theta)\), use de Moivre's theorem to deduce that
\(\sum_{m=1}^{\infty}\left(\frac{1}{2}\right)^{m} \sin m \theta=\frac{2 \sin \theta}{5-4 \cos \theta}\)

It is given that \(y=\ln (a x+1)\), where \(a\) is a positive constant. Prove by mathematical induction that, for every positive integer \(n\),
\(\frac{\mathrm{d}^{n} y}{\mathrm{~d} x^{n}}=(-1)^{n-1} \frac{(n-1)!a^{n}}{(a x+1)^{n}}\)

For the sequence \(u_{1}, u_{2}, u_{3}, \ldots\), it is given that \(u_{1}=8\) and
\(u_{r+1}=\frac{5 u_{r}-3}{4}\)
for all \(r\).
(i) Prove by mathematical induction that
\(u_{n}=4\left(\frac{5}{4}\right)^{n}+3,\)
for all positive integers \(n\).

(ii) Deduce the set of values of \(x\) for which the infinite series
\(\left(u_{1}-3\right) x+\left(u_{2}-3\right) x^{2}+\ldots+\left(u_{r}-3\right) x^{r}+\ldots\)
is convergent.

(iii) Use the result given in part (i) to find surds \(a\) and \(b\) such that
\(\sum_{n=1}^{N} \ln \left(u_{n}-3\right)=N^{2} \ln a+N \ln b\)

The sequence of positive numbers \(u_{1}, u_{2}, u_{3}, \ldots\) is such that \(u_{1}\lt 3\) and, for \(n \geqslant 1\),
\(u_{n+1}=\frac{4 u_{n}+9}{u_{n}+4} .\)
(i) By considering \(3-u_{n+1}\), or otherwise, prove by mathematical induction that \(u_{n}\lt 3\) for all positive integers \(n\).

(ii) Show that \(u_{n+1}\gt u_{n}\) for \(n \geqslant 1\).

Prove by mathematical induction that, for every positive integer \(n\),

\(\frac{d^{2n-1}}{dx^{2n-1}}(x\cos x)=(-1)^n\left(x\sin x-(2n-1)\cos x\right).\)

(a) The positive sequence \(u_1,u_2,u_3,\ldots\) satisfies \(u_1<5\) and \(u_{n+1}=\dfrac{6u_n+5}{u_n+2}\). By considering \(5-u_{n+1}\), prove by induction that \(u_n<5\) for all positive integers \(n\).

(b) Show that \(u_{n+1}>u_n\) for \(n\ge1\).

It is given that \(y=\mathrm{e}^{x} u\), where \(u\) is a function of \(x\). The \(r\) th derivatives \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{r}}\) and \(\frac{\mathrm{d}^{r} u}{\mathrm{~d} x^{r}}\) are denoted by \(y^{(r)}\) and \(u^{(r)}\) respectively. Prove by mathematical induction that, for all positive integers \(n\),
\(y^{(n)}=\mathrm{e}^{x}\left(\binom{n}{0} u+\binom{n}{1} u^{(1)}+\binom{n}{2} u^{(2)}+\ldots+\binom{n}{r} u^{(r)}+\ldots+\binom{n}{n} u^{(n)}\right) .\)

[You may use without proof the result \(\left.\binom{k}{r}+\binom{k}{r-1}=\binom{k+1}{r}.\right]\)

Prove, by mathematical induction, that \(\sum_{r=1}^{n} r \ln \left(\frac{r+1}{r}\right)=\ln \left(\frac{(n+1)^{n}}{n!}\right)\) for all positive integers \(n\).

Given that \(a\) is a constant, prove by mathematical induction that, for every positive integer \(n\),
\(\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(x \mathrm{e}^{a x}\right)=n a^{n-1} \mathrm{e}^{a x}+a^{n} x \mathrm{e}^{a x} .\)

Prove by mathematical induction that, for all positive integers \(n, \sum_{r=1}^{n} \frac{1}{(2 r)^{2}-1}=\frac{n}{2 n+1}\).

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

The sequence \(a_{1}, a_{2}, a_{3}, \ldots\) is such that \(a_{1}\gt 5\) and \(a_{n+1}=\frac{4 a_{n}}{5}+\frac{5}{a_{n}}\) for every positive integer \(n\). Prove by mathematical induction that \(a_{n}\gt 5\) for every positive integer \(n\).

Prove also that \(a_{n}\gt a_{n+1}\) for every positive integer \(n\).

Using factorials, show that \(\binom{n}{r-1}+\binom{n}{r}=\binom{n+1}{r}\).

Hence prove by mathematical induction that
\((a+x)^{n}=\binom{n}{0} a^{n}+\binom{n}{1} a^{n-1} x+\ldots+\binom{n}{r} a^{n-r} x^{r}+\ldots+\binom{n}{n} x^{n}\)
for every positive integer \(n\).

It is given that a diagonal of a polygon is a line joining two non-adjacent vertices. Prove, by mathematical induction, that an \(n\)-sided polygon has \(\frac{1}{2} n(n-3)\) diagonals, where \(n \geqslant 3\).

(i) Show that \(\frac{\mathrm{d}^{n+1}}{\mathrm{~d} x^{n+1}}\left(x^{n+1} \ln x\right)=\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(x^{n}+(n+1) x^{n} \ln x\right)\).

(ii) Prove by mathematical induction that, for all positive integers \(n\),
\(\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(x^{n} \ln x\right)=n!\left(\ln x+1+\frac{1}{2}+\ldots+\frac{1}{n}\right)\)

It is given that \(u_{r}=r \times r!\) for \(r=1,2,3, \ldots\). Let \(S_{n}=u_{1}+u_{2}+u_{3}+\ldots+u_{n}\). Write down the values of
\(2!-S_{1}, \quad 3!-S_{2}, \quad 4!-S_{3}, \quad 5!-S_{4} .\)

Conjecture a formula for \(S_{n}\).

Prove, by mathematical induction, a formula for \(S_{n}\), for all positive integers \(n\).

It is given that \(u_{r}=r \times r!\) for \(r=1,2,3, \ldots\). Let \(S_{n}=u_{1}+u_{2}+u_{3}+\ldots+u_{n}\). Write down the values of
\(2!-S_{1}, \quad 3!-S_{2}, \quad 4!-S_{3}, \quad 5!-S_{4} .\)

Conjecture a formula for \(S_{n}\).

Prove, by mathematical induction, a formula for \(S_{n}\), for all positive integers \(n\).

Prove by mathematical induction that, for every positive integer \(n\),
\(\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(\mathrm{e}^{x} \sin x\right)=(\sqrt{2})^{n} \mathrm{e}^{x} \sin \left(x+\frac{1}{4} n \pi\right) .\)

It is given that \(y=(1+x)^{2} \ln (1+x)\). Find \(\frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}\).

Prove by mathematical induction that, for every integer \(n \geqslant 3\),
\(\frac{\mathrm{d}^{n} y}{\mathrm{~d} x^{n}}=(-1)^{n-1} \frac{2(n-3)!}{(1+x)^{n-2}}\)

It is given that \(y=(1+x)^{2} \ln (1+x)\). Find \(\frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}\).

Prove by mathematical induction that, for every integer \(n \geqslant 3\),
\(\frac{\mathrm{d}^{n} y}{\mathrm{~d} x^{n}}=(-1)^{n-1} \frac{2(n-3)!}{(1+x)^{n-2}}\)

Prove by mathematical induction that, for every positive integer \(n\),
\((\cos \theta+\mathrm{i} \sin \theta)^{n}=\cos n \theta+\mathrm{i} \sin n \theta .\)

Express \(\sin ^{5} \theta\) in the form \(p \sin 5 \theta+q \sin 3 \theta+r \sin \theta\), where \(p, q\) and \(r\) are rational numbers to be determined.

Prove by induction that
\(\sum_{r=1}^{n}\left(3 r^{5}+r^{3}\right)=\frac{1}{2} n^{3}(n+1)^{3}\)
for all \(n \geqslant 1\).

Use this result together with the List of Formulae (MF10) to prove that
\(\sum_{r=1}^{n} r^{5}=\frac{1}{12} n^{2}(n+1)^{2} \mathrm{Q}(n),\)
where \(\mathrm{Q}(n)\) is a quadratic function of \(n\) which is to be determined.

For the sequence \(u_{1}, u_{2}, u_{3}, \ldots\), it is given that \(u_{1}=1\) and \(u_{r+1}=\frac{3 u_{r}-2}{4}\) for all \(r\). Prove by mathematical induction that \(u_{n}=4\left(\frac{3}{4}\right)^{n}-2\), for all positive integers \(n\).

Let \(S_{N}=\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\ldots+\frac{N}{(N+1)!}\). Prove by mathematical induction that, for all positive integers \(N\),
\(S_{N}=1-\frac{1}{(N+1)!} .\)

Let \(\mathbf{A}=\left(\begin{array}{ll}2 & 3 \\ 0 & 1\end{array}\right)\). Prove by mathematical induction that, for every positive integer \(n\),
\(\mathbf{A}^{n}=\left(\begin{array}{cc} 2^{n} & 3\left(2^{n}-1\right) \\ 0 & 1 \end{array}\right) .\)

Prove by mathematical induction that, for all positive integers \(n\),
\(\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(\mathrm{e}^{x} \sin x\right)=2^{\frac{1}{2} n} \mathrm{e}^{x} \sin \left(x+\frac{1}{4} n \pi\right) .\)

Prove by mathematical induction that, for all positive integers \(n\),
\(\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(\frac{1}{2 x+3}\right)=(-1)^{n} \frac{n!2^{n}}{(2 x+3)^{n+1}}\)

The sequence \(x_{1}, x_{2}, x_{3}, \ldots\) is such that \(x_{1}=3\) and
\(x_{n+1}=\frac{2 x_{n}^{2}+4 x_{n}-2}{2 x_{n}+3}\)
for \(n=1,2,3, \ldots\). Prove by induction that \(x_{n}\gt 2\) for all \(n\).