Proof by induction

Sample questions

Original practice questions written in the style of the exam.

1. Question 15 marks

   Prove by induction that, for all $n \in \mathbb{Z}^+$, $\begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}^n = \begin{pmatrix} 2^n & 2^n - 1 \\ 0 & 1 \end{pmatrix}$

   Mark scheme

   • B1 $n = 1$: right-hand side is $\begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}$, so true for (1)
2. Question 25 marks

   Prove by induction that $f(n) = 3^{2n} + 7$ is divisible by $8$ for all $n \in \mathbb{Z}^+$.