2017 RVHS Promo Q5

A sequence \({{U}_{1}}\), \({{U}_{2}}\), \({{U}_{3}}\),… is defined by \({{U}_{n}}=\frac{n}{{{\mathrm{e}}^{n}}}\).

1. Show that \(\frac{n\left( 1-\mathrm{e} \right)+1}{{{\mathrm{e}}^{n+1}}}={{U}_{n+1}}-{{U}_{n}}\) .[1]
2. Hence find \(\sum\limits_{r=1}^{n}{\left( \frac{r\left( 1-\mathrm{e} \right)+1}{{{\mathrm{e}}^{r+1}}} \right)}\) in terms of \(n\).[3]
3. Using result in part (ii), find \(\sum\limits_{r=5}^{20}{\left( \frac{\left( r-1 \right)\left( 1-\mathrm{e} \right)+1}{{{\mathrm{e}}^{r}}} \right)}\), expressing your answer as a single fraction in terms of \(\mathrm{e}\).[3]