Induction for divisibility and matrix powersAQA A-Level Further Maths: Mind map

Exam questions on Induction for divisibility and matrix powers

1. Let $f(n)=5^n+3$ for positive integers $n$. It is to be proved by induction that $f(n)$ is divisible by $4$ for all $n$.
   Assuming that $f(k)$ is divisible by $4$, show that $f(k+1)$ is divisible by $4$.2 marks
2. The matrix $\mathbf{M}=\begin{pmatrix}1&2\\ 0&1\end{pmatrix}$ satisfies $\mathbf{M}^n=\begin{pmatrix}1&2n\\ 0&1\end{pmatrix}$ for every positive integer $n$.
   Find the smallest positive integer $n$ for which the top-right entry of $\mathbf{M}^n$ is greater than $1000$.2 marks
3. Let $\mathbf{A}=\begin{pmatrix}3&-2\\ 2&-1\end{pmatrix}$. A student conjectures that $\mathbf{A}^n=\begin{pmatrix}1+2n&-2n\\ 2n&1-2n\end{pmatrix}$ for all positive integers $n$.
   Calculate $\mathbf{A}^2$ and $\mathbf{A}^3$, and show that the conjecture is correct for $n=2$ and $n=3$.3 marks