Further Mathematics 7366: Proof Topical Past Paper Questions

Prove by induction that, for all $ n \in \mathbb{N} $, the expression $$ 55^{n}=2^{n} $$ is divisible by 3 [4 marks] ___

(a) Prove by induction that, for all integers $ n \geq 1 $, $$\sum_{r=1}^{n} r^2 = \frac{1}{6} n (n + 1) (2n + 1)$$ [4 marks] Hence, or otherwise, write down a factorised expression for the sum of the first $ 2n $ squares 1^{2}+2^{2}+3^{2}+\cdots+(2n)^{2} [1 mark] _____ Use the f…

$ ^{2} $rove by induction that, for all integers $ n \geq 1 $, $$ (\mathbf{A}\mathbf{B}\mathbf{A}^{-1})^{n}=\mathbf{A}\mathbf{B}^{n}\mathbf{A}^{-1} $$ where A and B are square matrices of equal dimensions, and A is non-singular. [4 marks]