Mathematics A, Further H235: Proof Topical Past Paper Questions

You are given that $ \mathbf{A} = \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} $ where $ a $ is a constant. Prove by induction that $ \mathbf{A}^n = \begin{pmatrix} 1 & an \\ 0 & 1 \end{pmatrix} $ for all integers $ n \geqslant 1 $. [5]

(a) Express as a decimal (base-10) number the base-23 number $ 7119_{23} $. (b) Solve the linear congruence $ 7n+11 \equiv 9(\mod 23) $. [3] [3] (c) Let $N = 10a + b$ and $M = a + 7b$, where $a$ and $b$ are integers and $0 \leq b \leq 9$. (i) By considering $3N-7M$, prove that $2…

Prove that $ 2^{3n} - 3^n $ is divisible by 5 for all integers $ n \geqslant 1 $. [5]