2017 VJC P1 Q8

It is given that \(\sum\limits_{r=1}^{n}{\frac{{{r}^{2}}}{{{3}^{r}}}}=\frac{3}{2}-\frac{{{n}^{2}}+3n+3}{2\left( {{3}^{n}} \right)}\) .

1. Find \(\sum\limits_{r=1}^{\infty }{\frac{{{r}^{2}}+{{\left( -1 \right)}^{r}}}{{{3}^{r}}}}\).[3]
2. Show that \(\sum\limits_{r=4}^{n}{\frac{{{\left( r-2 \right)}^{2}}}{{{3}^{r-2}}}}=\frac{p}{q}-\frac{a{{n}^{2}}-an+a}{2\left( {{3}^{n-2}} \right)}\), where \(a\), \(p\) and \(q\) are integers to be determined.[5]