Proof by mathematical inductionEdexcel A-Level Further Maths: Mind map

Exam questions on Proof by mathematical induction

1. Proof by mathematical induction is used to show that $\sum_{r=1}^{n}\frac{1}{r(r+1)}=\frac{n}{n+1}$ for all positive integers $n$.
   Show that if the result is true for $n=k$ then it is true for $n=k+1$.2 marks
2. Let $\mathrm{f}(n)=7^n+5$, where $n$ is a positive integer.
   Given that $\mathrm{f}(k)$ is divisible by 6 for some positive integer $k$, show that $\mathrm{f}(k+1)$ is also divisible by 6.2 marks
3. The matrix $\mathbf{A}=\begin{pmatrix}2&1\\0&1\end{pmatrix}$.
   Find $\mathbf{A}^2$ and $\mathbf{A}^3$, and hence suggest a formula for $\mathbf{A}^n$.3 marks