It is given that $\sum\limits_{r=1}^{n}{\frac{{{r}^{2}}}{{{3}^{r}}}}=\frac{3}{2}-\frac{{{n}^{2}}+3n+

Question

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

Accepted Answer

(i) $\frac{5}{4}$ (ii) $p = 7, q = 6, a = 1$

2017 VJC P1 Q8

2017 VJC P1 Q8

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