2017 RVHS Promo Q5
A sequence ${{U}_{1}}$, ${{U}_{2}}$, ${{U}_{3}}$,… is defined by ${{U}_{n}}=\frac{n}{{{\text{e}}^{n}}}$.
(i)
Show that $\frac{n\left( 1-\text{e} \right)+1}{{{\text{e}}^{n+1}}}={{U}_{n+1}}-{{U}_{n}}$ .
[1]
(ii)
Hence find $\sum\limits_{r=1}^{n}{\left( \frac{r\left( 1-\text{e} \right)+1}{{{\text{e}}^{r+1}}} \right)}$ in terms of $n$.
[3]
(iii)
Using result in part (ii), find $\sum\limits_{r=5}^{20}{\left( \frac{\left( r-1 \right)\left( 1-\text{e} \right)+1}{{{\text{e}}^{r}}} \right)}$, expressing your answer as a single fraction in terms of $\text{e}$.
[3]
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