Consider two consecutive positr+u)x4z. w o.*xgk1t;gdj irgive 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 is divisible by xx qdd+3k0wl7 3 .

Prove that the sum of thrr*,:w20m c ;jar k24v zddravaee consecutive positive integers is divisible by 3 .

The product of three consecutive integers is increased by the middle incml/w vvm: q/rb gz/))teger. Prove that the rez)vm /vbgc//qw: lr m)sult is a perfect cube.

1. Show that $(2 n-1)^{3}+(2 n+1)^{3}=16 n^{3}+12 n$ 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, proyk4nhiq zy1 gr //n kjo*7 .tdyp*2r+tve that $ 1^{2}+2^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$ for all $ n \in \mathbb{Z}^{+} $.

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

Using the method of proof by contradictgnv9 w pz:- b1..kcp6rmwk;onion, prove that $ \sqrt{3} $ is irrational.

The Fibonacci sequence is defined as vf-za s k :agt*-axv98uz h62lfollows:

$\begin{array}{l}
a_{0}=0, a_{1}=1, a_{2}=1 \\
a_{n}=a_{n-1}+a_{n-2} \text { for } n \geq 2 .(F S)
\end{array}$

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 contradictioz6qd)1/tfx y kn that the equation $ 3 x^{3}-7 x^{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

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

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

Using the principle of ma2zs(mqhfi7ak k8 qg0 e+yc5m/thematical induction, prove that $n\left(n^{2}+5\right)$ is divisible by 6 \ for all integers $n \geq 1 \ $.

1. Solve the inequalk34e opg 3mh3aity $x^{2} \geq 2 x+3 $.
2. Use mathematical induction to prove that $2^{n}>n^{2}-2 $ for all $n \in \mathbb{Z}^{+}$, $n \geq 3$ .

1. Show that $\frac{1}{2 \sqrt{n+1}}<\sqrt{n+1}-\sqrt{n}$ , where $ \in \mathbb{Z}$, $n \geq 0$ .
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 \quad $ for all $ n \in \mathbb{Z}^{+}, n \geq 2$

This question asks you to investigax.h:c2wcz(eqvp;byqz7 7l,vly )u * ute some properties of hexagonal numbers. Hexagonal numbers can be represented by dots as shown below where7zqvu.y7bx; )vpqhucyz l *c ,:(l2 we $h_{n}$ 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 $h_{n+1}=h_{n}+4 n+1, n \in \mathbb{N}$ .
3. By expressing $h_{n}$ as a series, show that $h_{n}=2 n^{2}-n, n \in \mathbb{N}$ .
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}+4 n+1$, n $\in \mathbb{N}$ , then

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

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 $\frac{\mathrm{d}^{n} f}{\mathrm{~d} x^{n}}=\left(\frac{3 n+x-1}{3^{n}}\right) e^{\frac{x}{3}}$ 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 th1ov;pkau )w7h7xx oguo1 o t54eorem to find the value of $ \left[\cos \left(\frac{\pi}{6}\right)+\mathrm{i} \sin \left(\frac{\pi}{6}\right)\right]^{12}$ .
2. Use mathematical induction to prove that

$(\cos \alpha-\mathrm{i} \sin \alpha)^{n}=\cos (n \alpha)-\mathrm{i} \sin (n \alpha) \quad $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^{*}\right)^{n}$, $n \in \mathbb{Z}^{+}$ , where $ w^{*}$ is the complex conjugate of w .
4. 1. Show that $w w^{*}=1$ .
2. Write down and simplify the binomial expansion of $ \left(w-w^{*}\right)^{3}$ in terms of w and $w^{*}$ .
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 $0 \leq \alpha \leq \pi$ .

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

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

Let $g(x)=\cos (m x), 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 opp7iojz9wak(as0/-.t/ cu dthe graph of $y=\arctan (2 x-3)+\frac{3 \pi}{4} for x \in \mathbb{R}$ , with asymptotes at $y=\frac{\pi}{4} and y=\frac{5 \pi}{4}$ .



1. Describe a sequence of transformations that transforms the graph of $y=\arctan x$ to the graph of $y=\arctan (2 x-3)+\frac{3 \pi}{4}$ for $ \in \mathbb{R} $.
2. Show that $\arctan p-\arctan q \equiv \arctan \left(\frac{p-q}{1+p q}\right)$.
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} \quad $ for $n \in \mathbb{Z}^{+}$