Further Pure Mathematics WFM01: PROOF BY INDUCTION Topical Past Paper Questions

A sequence of numbers $u_{1}, u_{2}, u_{3}, \ldots$ is defined by $$u_{1} = 8 \quad u_{2} = 10$$ $$u_{n+2} = u_{n+1} + 2u_{n} \quad n \in \mathbb{Z}^{+}$$ Prove by induction that, for $n \in \mathbb{Z}^{+}$ $$u_{n} = 3 \times 2^{n} - 2 \times (-1)^{n}$$ [6]

(a) Using the standard summation formulae show that $$ \sum_{r=1}^{2n}\left(2r^{2}-1\right)=\frac{4}{3}n(n+1)(an+b) $$ where a and b are integers to be determined. (5) (b) Prove by induction that, for $ n \in \mathbb{N} $ $$ \sum_{r=1}^{n}r(3r-2)^{2}=\frac{n^{2}(n+1)(9n-7)}{4} $$…

Prove by induction that for $ n \in \mathbb{Z}^+ $ $$ \mathrm{f}(n)=7^{n-1}+8^{2n+1} $$ is divisible by 57 (6)