Proofs

[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: 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: 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: 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]