2020 RI P1 Q7

It is given that \(\mathrm{f}(r)=\frac{{{2}^{r}}}{r-2}\). Show that \(\mathrm{f}(r+2)-\mathrm{f}(r)=\frac{\left( 3r-8 \right){{2}^{r}}}{r(r-2)}\).[2]

1. Show that \(\sum\limits_{r=3}^{n}{\frac{(3r-8){{2}^{r}}}{r(r-2)}}=\frac{(3n-2){{2}^{n+1}}}{n(n-1)}-16\).[4]
2. Hence find \(\sum\limits_{r=1}^{n}{\frac{(3r-2){{2}^{r}}}{r(r+2)}}\) in the form \(\frac{(3n+A){{2}^{n+1}}}{(n+B)(n+1)}+C\) where \(A\),\(B\) and \(C\) are integers to be determined.[4]