Explanation

(a)
$\sum_{r = 0}^{n}{( - 1)^{r}\ ^{n}C_{r}}\left( \frac{1}{2^{r}} + \frac{3^{r}}{2^{2r}} + \frac{7^{r}}{2^{3r}} + ...upto\ m\ terms \right)$
$= \sum_{r = 0}^{n}{{( - 1)}^{r}\ ^{n}C_{r} \bullet \frac{1}{2^{r}} + \sum_{r = 0}^{n}{{( - 1)}^{r} \bullet \ ^{n}C_{r}\frac{3^{r}}{2^{2r}} + \sum_{r = 0}^{n}{( - 1)^{r}\ ^{n}C_{r}\frac{7^{r}}{2^{3r}} + ...}}}$
$= \left( 1 - \frac{1}{2} \right)^{n} + \left( 1 - \frac{3}{4} \right)^{n} + \left( 1 - \frac{7}{8} \right)^{n} + ...upto\ m\ terms$
$= \frac{1}{2^{n}} + \frac{1}{2^{2n}} + \frac{1}{2^{3n}} + \ldots upto\ m\ terms$
$= \frac{\frac{1}{2^{n}}\left( 1 - \left( \frac{1}{2^{n}} \right)^{m} \right)}{\left( 1 - \frac{1}{2^{n}} \right)}$
$= \frac{2^{mn} - 1}{2^{mn}(2^{n} - 1)}$