Further Mathematics 7367: Proof Topical Past Paper Questions

The sequence $u_{1}, u_{2}, u_{3}, \ldots$ is defined by $$ u_{1}=1 $$ $$ u_{n+1}=u_{n}+3n $$ Prove by induction that for all integers $ n \geq 1 $ $$ u_{n}=\frac{3}{2}n^{2}-\frac{3}{2}n+1 $$

The function f is defined by $$ \mathrm{f}(n)=3^{3n+1}+2^{3n+4}\qquad(n\in\mathbb{Z}^{+}) $$ Prove by induction that $ f(n) $ is divisible by 19 for $ n \geq 1 $ [6 marks]

Prove by induction that, for all integers $ n \geq 1 $, $$\sum_{r=1}^{n} r^{3} = \left\{ \frac{1}{2} n (n + 1) \right\}^{4}$$ [4 marks]