Proofs

[Maximum mark: 4]

Prove that the sum of three consecutive positive integers is divisible by $3$.

[Maximum mark: 4]

Consider two consecutive positive integers, $k$ and $k+1$.

Show that the difference of their squares is equal to the sum of the two integers.

[Maximum mark: 4]

The product of three consecutive integers is increased by the middle integer.

Prove that the result is a perfect cube.

[Maximum mark: 6]

1. Show that $(2n-1)^3 + (2n+1)^3 = 16n^3+12n$ for $n \in \mathbb{Z}$. [3]
   

2. Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four. [3]
   

[Maximum mark: 6]

Using mathematical induction, prove that $1^2 + 2^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$ for all $n \in \mathbb{Z}^+$.

[Maximum mark: 5]

1. Prove that $\dfrac{5}{x^2} = \dfrac{5}{x(x-2)}-\dfrac{10}{x^2(x-2)}$. [3]
   

2. Determine the set of numbers $x$ for which the proof in part (a) is valid. [2]
   

[Maximum mark: 7]

Use the principle of mathematical induction to prove that

[Maximum mark: 4]

Using the method of proof by contradiction, prove that $\sqrt{3}$ is irrational.

[Maximum mark: 6]

Let $r \in \mathbb{R}, r\neq 1$. Use the method of mathematical induction to prove that

[Maximum mark: 6]

Prove by contradiction that the equation $3x^3-7x^2+5=0$ has no integer roots.

[Maximum mark: 7]

The Fibonacci sequence is defined as follows:

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

[Maximum mark: 8]

Let $y = x^2 e^x$, for $x \in \mathbb{R}$.

1. Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$. [1]
   

2. Prove by mathematical induction that

   $$\hspace{4em} \dfrac{\mathrm{d}^n}{\mathrm{d}x^n}\big(x^2e^x\big) = \big(n(n-1) + 2nx + x^2\big)e^x \hspace{1.5em} \text{for all $n \in \mathbb{Z}^+$, $n\geq2$.}$$

   [7]
   

[Maximum mark: 9]

Let $f(x) = (x+1)e^{-2x}$, $x \in \mathbb{R}$.

1. Find $f'(x)$. [2]
   

2. Prove by induction that $\dfrac{\mathrm{d}^nf}{\mathrm{d}x^n} = \big[n(-2)^{n-1} + (-2)^n(x+1)\big]e^{-2x}$ for all $n \in \mathbb{Z}^+$.[7]
   

[Maximum mark: 6]

Using the principle of mathematical induction, prove that $n(n^2+5)$ is divisible by $6$ for all integers $n \geq 1$.

[Maximum mark: 8]

1. Solve the inequality $x^2 \geq 2x + 3$. [2]
   

2. Use mathematical induction to prove that $2^n > n^2 - 2$ for all $n \in \mathbb{Z}^+$, $n \geq 3$.[6]
   

[Maximum mark: 14]

1. Show that $\dfrac{1}{2\sqrt{n+1}} < \sqrt{n+1} - \sqrt{n}$, where $n \in \mathbb{Z},\hspace{0.1em} n\geq 0$. [3]
   

2. Hence show that $\dfrac{1}{\sqrt{2}} < 2\sqrt{2} - 2$. [2]
   

3. Prove by mathematical induction that

   $$\hspace{4em} \sum_{r = 2}^n \dfrac{1}{\sqrt{r}} < 2\sqrt{n} - 2 \hspace{2em} \text{for all $n \in \mathbb{Z}^+$, $n \geq 2$.}$$

   [9]
   

[Maximum mark: 28]

This question asks you to explore the sequence defined by

where $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2-4x+1=0, \, \alpha > \beta$ and $n \in \mathbb{Z}^+$.

1. Find the value of $\alpha$ and the value of $\beta$. Give your answers in the form $a \pm \sqrt{b}$, where $a,b \in \mathbb{Z^+}$.[3]
   

2. Hence find the values of $u_1$ and $u_2$. [4]
   

3. Show that $\alpha^2 = 4\alpha -1$ and $\beta^2 = 4\beta - 1$. [1]
   

4. Hence show that $u_{n+2} = 4u_{n+1}-u_n$.[4]
   

5. Suppose that $u_n$ and $u_{n+1}$ are integers. Show that $u_{n+2}$ is also an integer.[2]
   

6. Hence show that $u_n$ is an integer for all $n \in \mathbb{N}$.[2]
   

Now consider the sequence defined by

7. Find the exact values of $v_1$ and $v_2$.[4]
   

8. Express $v_{n+2}$ in terms of $v_{n+1}$ and $v_n$.[4]
   

9. Hence show that $v_n$ is a multiple of $\dfrac{\sqrt{3}}{3}$ for all $n \in \mathbb{N}$.[4]
   

[Maximum mark: 24]

This question asks you to investigate some properties of hexagonal numbers.

Hexagonal 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 $h_5$.[1]
   

2. By examining the pattern, show that $h_{n+1} = h_{n}+4n+1$, $n\in \mathbb{N}$. [3]
   

3. By expressing $h_n$ as a series, show that $h_n = 2n^2-n$, $n\in \mathbb{N}$.[3]
   

4. Hence, determine whether $2016$ is a hexagonal number.[3]
   

5. Find the least hexagonal number which is greater than $80\hspace{0.10em}000$.[5]
   

6. Consider the statement:

   $45$ is the only hexagonal number which is divisible by $9$.

   Show that this statement is false.[2]
   

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

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

[Maximum mark: 21]

1. Use de Moivre's theorem to find the value of $\left[\cos\left(\dfrac{\pi}{6}\right) + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \left(\dfrac{\pi}{6}\right)\right]^{12}$. [2]
   

2. Use mathematical induction to prove that

   $$\hspace{3.5em} (\cos \alpha - {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha)^n = \cos (n\alpha) - {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin (n\alpha) \hspace{1em} \text{for all } n \in \mathbb{Z}^+.$$

   [6]
   

Let $w = \cos \alpha + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha$.

3. Find an expression in terms of $\alpha$ for $w^n - (w^\ast)^n$, $n \in \mathbb{Z}^+$, where $w^\ast$ is the complex conjugate of $w$. [2]
   

4. 1. Show that $ww^\ast = 1$.

   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]
      

5. Hence solve $4\sin^3\alpha + (2 \cos \alpha - 3) \sin \alpha = 0$ for $0 \leq \alpha \leq \pi$. [6]
   

[Maximum mark: 23]

Let $f(x) = (x-1)e^{\frac{x}{3}}$, for $x \in \mathbb{R}$.

1. Find $f'(x)$. [2]
   

2. Prove by induction that $\dfrac{\mathrm{d}^nf}{\mathrm{d}x^n} = \bigg(\dfrac{3n + x - 1}{3^n}\bigg)e^{\frac{x}{3}}$ for all $n \in \mathbb{Z}^+$. [7]
   

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. [5]
   

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]
   

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. [4]
   

[Maximum mark: 21]

Let $f(x) = \dfrac{1}{\sqrt{1-x}}$,$x < 1$.

1. Show that $f''(x) = \dfrac{3}{4} (1-x)^{-5/2}$. [3]
   

2. Use mathematical induction to prove that[9]
   

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

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 $-\dfrac{3}{4}$.

3. Find the possible values of $m$.[9]
   

[Maximum mark: 17]

The following diagram shows the graph of $y=\arctan(2x-3)+\dfrac{3\pi}{4}$ for $x\in \mathbb{R}$,
with asymptotes at $y=\dfrac{\pi}{4}$ and $y=\dfrac{5\pi}{4}$.

1. Describe a sequence of transformations that transforms the graph of
   $y=\arctan x$ to the graph of $y=\arctan(2x-3)+\dfrac{3\pi}{4}$ for $x\in \mathbb{R}$.[3]
   

2. Show that $\arctan p - \arctan q \equiv \arctan \left(\dfrac{p-q}{1+pq}\right)$.[3]
   

3. Verify that $\arctan(x+2)-\arctan(x+1) = \arctan\left( \dfrac{1}{(x+1)^2+(x+1)+1}\right)$.[3]
   

4. Using mathematical induction and the results from part (b) and (c), prove that[8]
   

   $$\sum_{r=1}^n \arctan\left(\dfrac{1}{r^2+r+1}\right) = \arctan(n+1)-\dfrac{\pi}{4} \hspace{0.8em} \text{for } n\in \mathbb{Z}^{+}.$$