Explanation

(a)
Let $S = 1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16 + ...}$
$= 1 + \frac{(4 - 1)}{2} + \frac{(8 - 1)}{4} + \frac{(16 - 1)}{8} + \frac{(32 - 1)}{16} + ...$
$= 1 + 2 - \frac{1}{2} + 2 - \frac{1}{4} + 2 - \frac{1}{8} + 2 - \frac{1}{16} + ...$
$= 1 + 2(n - 1) - \left\lbrack \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... + (n - 1) \right\rbrack$
$1 + 2(n - 1) - \left\lbrack \frac{\frac{1}{2}\left( 1 - \frac{1}{2^{n - 1}} \right)}{1 - \frac{1}{2}} \right\rbrack$
$= 1 + 2(n - 1) - 1 + \frac{1}{2^{n - 1}} = 2(n - 1) + \frac{1}{2^{n - 1}}$