Consider two consecutive pi .;)s l+ eyyy wr*pm0sjc;st2ositive integers, k and k+1 .
Show that the difference of their squares is equal to the sum of the two integers.

Prove that the sum of three consecutive positive integers 3)q.jdsklv5 bbp ) 3nc// w-n 9dmqyi:tadj/his divisible by 3 .

Prove that the sum of thre f2.xy7isz n): wtk9kce consecutive positive integers is divisible by 3 .

The product of three consecutive integers is increap6 8m lojpi0g14jf)k1ag /jqz sed by the middle integer. Prove thatg4)gqj aji1l8oz k f061mpjp / the result is a perfect cube.

1. Show that $(2n-1{)}^3+(2n+1{)}^3=16n^3+12n$ for $n\in \mathbb{Z}$ .
2. Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four.

Using mathematical induction, prove that 0m:+1 t.hpndo *qiy gk$1^2+2^2+\cdots +n^2=\frac{n(n+1)(2n+1)}{6}$ for all $n\in {\mathbb{Z}}^{+}$.

Let $r\in \mathbb{R},r\ne$1 . Use the method of mathematical induction to prove that
$1+r+r^2+\cdots +r^n=\frac{1-r^{n+1}}{1-r}$ for all $n\in {\mathbb{Z}}^{+}$ .

Using the method of proof by contradiction, pn(5v298t lb .7ja88le7qqazhngug 2cv v b p0hrove that $\sqrt{3}$ is irrational.

The Fibonacci sequence flfur)z;y+;ds4v y3 gis defined as follows:

$\begin{array}{l}{a}_0=0,a_1=1,a_2=1\\ a_n=a_{n-1}+a_{n-2}\text{ for }n\ge 2.(FS)\end{array}$

Prove by mathematical induction that $a_1^2+a_2^2+\cdots +a_n^2=a_n{a}_{n+1}$ , where $n\in {\mathbb{Z}}^{+}$ .

Prove by contradiction that td 0zf/b (/v n*j1sshavhe equation $3x^3-7x^2+5=0$ has no integer roots.

Let $y=x^2{e}^x$ , for $x\in \mathbb{R}$ .
1. Find $frac\mathrm{d}y\mathrm{d}x$ .
2. Prove by mathematical induction that

$\frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}\left(x^2{e}^x\right)=(n(n-1)+2nx+x^2)e^x\text{ for all }n\in {\mathbb{Z}}^{+},n\ge 2$

Let $f(x)=(x+1)e^{-2x}$, $x\in \mathbb{R}$ .
1. Find $f^{\prime }(x)$ .
2. Prove by induction that $\frac{{\mathrm{d}}^{n}f}{\mathrm{d}{x}^n}=[n(-2{)}^{n-1}+(-2{)}^n(x+1)]e^{-2x}foralln\in {\mathbb{Z}}^{+}$ .

Using the principle of mathematical induction, p,yv+pr; bm5grrove that $n(n^2+5)$ is divisible by 6 \ for all integers $n\ge 1$.

1. Solve the inequalid hsrw37w k75sty $x^2\ge 2x+3$.
2. Use mathematical induction to prove that $2^n>n^2-2$ for all $n\in {\mathbb{Z}}^{+}$, $n\ge 3$ .

1. Show that $\frac{1}{2\sqrt{n+1}}<\sqrt{n+1}-\sqrt{n}$ , where $\in \mathbb{Z}$, $n\ge 0$ .
2. Hence show that $\frac{1}{\sqrt{2}}<2\sqrt{2}-2$ .
3. Prove by mathematical induction that

$\sum _{r=2}^n\frac{1}{\sqrt{r}}<2\sqrt{n}-2$ for all $n\in {\mathbb{Z}}^{+},n\ge 2$

This question asks you to investigate some properties of hexagonal 9 u/ -m,bajef0efw 2dfnumbers. Hexagonaluefad /ejb09f,wm f-2 numbers can be represented by dots as shown below where $h_n$ denotes the n th hexagonal number, $n\in \mathbb{N}$.



Note that 6 points are required to create the regular hexagon $h_2$ with side of length 1 , while 15 points are required to create the next hexagon $h_3$ with side of length 2 , and so on.
1. Write down the value of $h_5$ .
2. By examining the pattern, show that $h_{n+1}=h_n+4n+1,n\in \mathbb{N}$ .
3. By expressing $h_n$ as a series, show that $h_n=2n^2-n,n\in \mathbb{N}$ .
4. Hence, determine whether 2016 is a hexagonal number.
5. Find the least hexagonal number which is greater than 80000 .
6. Consider the statement:
45 is the only hexagonal number which is divisible by 9 .
Show that this statement is false.

Matt claims that given $h_1=1$ and $h_{n+1}=h_n+4n+1$, n $\in \mathbb{N}$ , then

$h_n=2n^2-n,n\in \mathbb{N}$

7. Show, by mathematical induction, that Matt's claim is true for all n $\in \mathbb{N}$ .

Let $f(x)=(x-1)e^{\frac{x}{3}}$ , for $x\in \mathbb{R}$ .
1. Find $f^{\prime }(x)$ .
2. Prove by induction that $\frac{{\mathrm{d}}^{n}f}{\mathrm{d}{x}^n}=\left(\frac{3n+x-1}{3^n}\right)e^{\frac{x}{3}}$ for all $n\in {\mathbb{Z}}^{+}$ .
3. Find the coordinates of any local maximum and minimum points on the graph of y=f(x) . Justify whether such point is a maximum or a minimum.
4. Find the coordinates of any points of inflexion on the graph of y=f(x) . Justify whether such point is a point of inflexion.
5. Hence sketch the graph of $y=f(x)$ , indicating clearly the points found in parts (c) and (d) and any intercepts with the axes.

1. Use de Moivre's thknzdnpv(tg2k 2ab ,(2w ,ao*yeorem to find the value of ${[\cos \left(\frac{\pi }{6}\right)+\mathrm{i}\sin \left(\frac{\pi }{6}\right)]}^{12}$ .
2. Use mathematical induction to prove that

$(\cos \alpha -\mathrm{i}\sin \alpha {)}^n=\cos (n\alpha )-\mathrm{i}\sin (n\alpha )$for all $n\in {\mathbb{Z}}^{+}$.

Let $w=\cos \alpha +\mathrm{i}\sin \alpha$ .
3. Find an expression in terms of $\alpha$ for $w^n-{\left(w^{\ast }\right)}^n$, $n\in {\mathbb{Z}}^{+}$ , where $w^{\ast }$ is the complex conjugate of w .
4. 1. Show that $w{w}^{\ast }=1$ .
2. Write down and simplify the binomial expansion of ${(w-w^{\ast })}^3$ in terms of w and $w^{\ast }$ .
3. Hence show that $\sin (3\alpha )=3\sin \alpha -4{\sin }^3\alpha$ .
5. Hence solve $4{\sin }^3\alpha +(2\cos \alpha -3)\sin \alpha =0$ for $0\le \alpha \le \pi$ .

Let $f(x)=\frac{1}{\sqrt{1-x}},x<1$.
1. Show that f${}^{\prime \prime }(x)=\frac{3}{4}(1-x{)}^{-5/2}$ .
2. Use mathematical induction to prove that

$f^{(n)}(x)={\left(\frac{1}{4}\right)}^n\frac{(2n)!}{n!}(1-x{)}^{-1/2-n}n\in \mathbb{Z}$, $n\ge 2$

Let $g(x)=\cos (mx),m\in \mathbb{Q}$.
Consider the function h defined by $h(x)=f(x)\times g(x)$ for x<1 .
The $x^2$ term in the Maclaurin series for h(x) has a coefficient of $-\frac{3}{4}$.
3. Find the possible values of m .

The following diagram shows the graph of l :0pimriow*5hc jz fs+ 88qoe 7-ew:j $y=\arctan (2x-3)+\frac{3\pi }{4}forx\in \mathbb{R}$ , with asymptotes at $y=\frac{\pi }{4}andy=\frac{5\pi }{4}$ .



1. Describe a sequence of transformations that transforms the graph of $y=\arctan x$ to the graph of $y=\arctan (2x-3)+\frac{3\pi }{4}$ for $\in \mathbb{R}$.
2. Show that $\arctan p-\arctan q\equiv \arctan \left(\frac{p-q}{1+pq}\right)$.
3. Verify that $\arctan (x+2)-\arctan (x+1)=\arctan \left(\frac{1}{(x+1{)}^2+(x+1)+1}\right)$ .
4. Using mathematical induction and the results from part (b) and (c), prove that

$\sum _{r=1}^n\arctan \left(\frac{1}{r^2+r+1}\right)=\arctan (n+1)-\frac{\pi }{4}$ for $n\in {\mathbb{Z}}^{+}$