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# Proofs

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Paper 2

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##### Question 1

no calculator

easy

[Maximum mark: 6]

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. 

easy

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##### Question 2

no calculator

easy

[Maximum mark: 4]

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

easy

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##### Question 3

no calculator

easy

[Maximum mark: 6]

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

\begin{aligned} \hspace{8.3em} 1+r+r^2+\cdots+r^n=\frac{1-r^{n+1}}{1-r} \hspace{2em} \text{for all } n\in \mathbb{Z}^+. \\ \end{aligned}

easy

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##### Question 4

no calculator

medium

[Maximum mark: 7]

The Fibonacci sequence is defined as follows:

\begin{aligned} a_0 &= 0,\hspace{0.25em} a_1 = 1,\hspace{0.25em} a_2 = 1, \\[6pt] a_n &= a_{n-1}+a_{n-2} \hspace{0.5em}\text{for}\hspace{0.5em} n \geq 2. \qref{(FS)}\end{aligned}

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

medium

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##### Question 5

no calculator

medium

[Maximum mark: 6]

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

medium

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##### Question 6

calculator

hard

[Maximum mark: 8]

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

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

hard

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##### Question 7

no calculator

hard

[Maximum mark: 14]

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

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

3. Prove by mathematical induction that

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



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