N2016 P1 Q6

Prove by the method of mathematical induction that \[\sum_{r=1}^n r(r^2+1)=\frac14n(n+1)(n^2+n+2).\][5]

A sequence \(u_0,u_1,u_2,\ldots\) is given by \[u_0=2\quad\text{and}\quad u_n=u_{n-1}+n^3+n\quad\text{for}\hspace{0.5em}n\ge1.\]

Find \(u_1\), \(u_2\) and \(u_3\).[2]

By considering \(\displaystyle\sum_{r=1}^n(u_r-u_{r-1})\), find a formula for \(u_n\) in terms of \(n\).[3]