2019 ACJC Promo Q10

1. It is given that \(\mathrm{f}\left( r \right)=r{{\left( \frac{1}{2} \right)}^{r}}\). Show that \(\mathrm{f}\left( r \right)-\mathrm{f}\left( r-1 \right)={{\left( \frac{1}{2} \right)}^{r-1}}-r{{\left( \frac{1}{2} \right)}^{r}}\).[1]
2. Hence use the method of differences to show
   \(\sum\limits_{r=1}^{n}{r{{\left( \frac{1}{2} \right)}^{r}}}=2-{{\left( \frac{1}{2} \right)}^{n-1}}-n{{\left( \frac{1}{2} \right)}^{n}}\).[4]

The function \(\mathrm{g}\), with domain the set of positive integers, is given by \(g\left( n \right)=\sum\limits_{r=1}^{n}{r{{\left( \frac{1}{2} \right)}^{r}}}\).

3. If \({{n}_{1}}\) and \({{n}_{2}}\) are two positive integers such that \({{n}_{2}}>{{n}_{1}}\), by considering \(g\left( {{n}_{2}} \right)-g\left( {{n}_{1}} \right)\), explain clearly why \(\mathrm{g}\) has an inverse and find \({{g}^{-1}}\left( \frac{251}{128} \right)\).[4]