It is given that $\mathrm{f}\left( r \right)=\left( r-2 \right){{2}^{r-1}}$. Show that \

Question

It is given that $\mathrm{f}\left( r \right)=\left( r-2 \right){{2}^{r-1}}$. Show that $\mathrm{f}\left( r+1 \right)-\mathrm{f}\left( r \right)={{2}^{r}}ar$, where $a$ is a constant to be determined.[2] Hence find $\sum\limits_{r=1}^{n}{r{{2}^{r}}}$.[3] Using the result in part (ii), find $\sum\limits_{r=3}^{n}{\left( r+2 \right){{2}^{r}}}$ in terms of $n$.[3]

Accepted Answer

(i) $a=\frac12$ (ii) $\sum_{r=1}^n r2^r=2(n-1)2^n+2$ (iii) $\sum_{r=3}^n(r+1)2^r=2^{n+1}(n+1)-24$

2020 HCI P2 Q3

2020 HCI P2 Q3

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