Prove by the method of mathematical induction that \[\sum_{r=1}^n r(2r^2+1)=\frac12n(n+1)(n^2+n+1).\][5]
It is given that \(f(r)=2r^3+3r^2+r+24\). Show that \(f(r)-f(r-1)=ar^2\), for a constant \(a\) to be determined. Hence find a formula for \(\displaystyle\sum_{r=1}^n r^2\), fully factorising your answer.[5]
Find \(\displaystyle\sum_{r=1}^n f(r)\). (You should not simplify your answer.)[3]