Use the principle of mathematical induction to prove that

$\sin x+\sin 3x+\dots +\sin (2n-1)x=\frac{1-\cos 2nx}{2\sin x},n\in {\mathbb{Z}}^{+},x\ne k\pi$ where $k\in \mathbb{Z}$.

Prove by contradiction that the equation $2x^3+6x+1=0$ has no integer roots.

Prove by mathematical induction that $\left(\begin{array}{c}2\\ 2\end{array}\right)+\left(\begin{array}{c}3\\ 2\end{array}\right)+\left(\begin{array}{c}4\\ 2\end{array}\right)+\dots +\left(\begin{array}{c}n-1\\ 2\end{array}\right)=\left(\begin{array}{c}n\\ 3\end{array}\right)$, where $n\in \mathbb{Z},n⩾3$.

Use mathematical induction to prove that $\sum _{r=1}^{n}r\left(r\text{!}\right)=\left(n+1\right)\text{!}-1$, for $n\in {\mathbb{Z}}^{+}$.

Hence or otherwise, find an expression for the derivative of $f_n(x)$ with respect to $x$.

Hence or otherwise, determine the value of $\underset{x\to 0}{\lim }\left[\frac{{\left(x^2{\mathrm{e}}^x-x^2\right)}^3}{x^9}\right]$.

Use the principle of mathematical induction to prove that

$1+2\left(\frac{1}{2}\right)+3{\left(\frac{1}{2}\right)}^2+4{\left(\frac{1}{2}\right)}^3+\dots +n{\left(\frac{1}{2}\right)}^{n-1}=4-\frac{n+2}{2^{n-1}}$, where $n\in {\mathbb{Z}}^{+}$.

Consider the function $f\left(x\right)=x{\text{e}}^{2x}$, where $x\in \mathbb{R}$. The $n^{\text{th}}$ derivative of $f\left(x\right)$ is denoted by $f^{\left(n\right)}\left(x\right)$.

Prove, by mathematical induction, that $f^{\left(n\right)}\left(x\right)=\left(2^{n}x+n{2}^{n-1}\right){\text{e}}^{2x}$, $n\in {\mathbb{Z}}^{+}$.

By using mathematical induction, prove that

$f_n(x)=\frac{\sin 2^{n+1}x}{2^n\sin 2x},x\ne \frac{m\pi }{2}$ where $m\in \mathbb{Z}$.

A polygonal number, $P_r\left(n\right)$, can be represented by the series

$\underset{m=1}{\overset{n}{\text{Σ}}}\left(1+\left(m-1\right)\left(r-2\right)\right)$ where $r\in {\mathrm{\mathbb{Z}}}^{+}, r\ge 3$.

Use mathematical induction to prove that $P_r\left(n\right)=\frac{\left(r-2\right)n^2-\left(r-4\right)n}{2}$ where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Hence state a condition in terms of $p$ and $q$ that would imply $x^4+p{x}^3+q{x}^2+rx+s=0$ has at least one complex root.

It is given that $2 \cos A \sin B\equiv \sin \left(A+B\right)-\sin \left(A-B\right)$. (Do not prove this identity.)

Using mathematical induction and the above identity, prove that $\underset{r=1}{\overset{n}{\mathrm{\Sigma }}}\cos \left(2r-1\right)\theta =\frac{\sin {\displaystyle }{\displaystyle 2}{\displaystyle n}{\displaystyle \theta }}{{\displaystyle 2}{\displaystyle }\sin \theta }$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Use mathematical induction to prove that $\frac{{\text{d}}^n}{\text{d}{x}^n}\left(x{\text{e}}^{px}\right)=p^{n-1}\left(px+n\right){\text{e}}^{px}$ for $n \in {\mathrm{\mathbb{Z}}}^{+}, p \in \mathrm{\mathbb{Q}}$.

Use mathematical induction to prove that $f^{\left(n\right)}\left(x\right)={\left(-\frac{1}{4}\right)}^{n-1}\frac{\left(2n-3\right)!}{\left(n-2\right)!}{\left(1+x\right)}^{\frac{1}{2}-n}$ for $n\in \mathrm{\mathbb{Z}}, n\ge 2$.

Prove by contradiction that ${\log }_2 5$ is an irrational number.

Hence or otherwise solve the equation $\sin x+\sin 3x=\cos x$ in the interval .

Use proof by contradiction to prove that a prime number, $p$, that is not of the form $a^2+b^2$ is a Gaussian prime.

Consider integers $a$ and $b$ such that $a^2+b^2$ is exactly divisible by $4$. Prove by contradiction that $a$ and $b$ cannot both be odd.

Given that $p^2<3q$, deduce that $\alpha , \beta$ and $\gamma$ cannot all be real.

Prove by mathematical induction that $\frac{d^n}{d{x}^n}\left(x^2{\mathrm{e}}^x\right)=\left[x^2+2nx+n\left(n-1\right)\right]{\mathrm{e}}^x$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Solve the inequality $x^2>2x+1$.

Determine whether $f_n$ is an odd or even function, justifying your answer.

Find the value of $\sin \frac{\pi }{4}+\sin \frac{3\pi }{4}+\sin \frac{5\pi }{4}+\sin \frac{7\pi }{4}+\sin \frac{9\pi }{4}$.

Use mathematical induction to prove that $2^{n+1}>n^2$ for $n\in \mathbb{Z}$, $n⩾3$.

Show that, for $n>1$, the equation of the tangent to the curve $y=f_n(x)$ at $x=\frac{\pi }{4}$ is $4x-2y-\pi =0$.

Use the method of mathematical induction to prove that $4^n+15n-1$ is divisible by 9 for $n\in {\mathbb{Z}}^{+}$.

Using mathematical induction and the result from part (b), prove that $\underset{r=1}{\overset{n}{\text{Σ}}}\text{arctan}\left(\frac{1}{2r^2}\right)=\text{arctan}\left(\frac{n}{n+1}\right)$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Hence or otherwise, determine the Maclaurin series of $f\left(x\right)=x^2{\mathrm{e}}^x$ in ascending powers of $x$, up to and including the term in $x^4$.