Proof by induction – further maths test to download as PDF

Prove by induction that \(\displaystyle\sum_{r=1}^{n}r(r+2)=\tfrac16n(n+1)(2n+7)\) for all positive integers \(n\).

Prove by induction that \(6^n+4\) is divisible by 5 for all positive integers \(n\).

Prove by induction that \(\begin{pmatrix}3&-4\\1&-1\end{pmatrix}^n=\begin{pmatrix}1+2n&-4n\\n&1-2n\end{pmatrix}\) for all \(n\in\mathbb Z^+\).

A sequence has \(u_1=5\) and \(u_{n+1}=4u_n-3\).

1. Find \(u_2\) and \(u_3\). [1]
2. Prove by induction that \(u_n=4^n+1\). [3]

After \(n\) weeks, account A has \(2^n\) followers and account B has \(n^2\) followers.

1. Show that the accounts have equal numbers of followers when \(n=4\), and that A has more when \(n=5\). [1]
2. Prove by induction that \(2^n>n^2\) for all \(n\ge5\). You may use the fact that \(2k^2>(k+1)^2\) for \(k\ge5\). [3]