Explanation

(b)
$S = 1 + 3\alpha + 5\alpha^{2} + ... + (2n - 1)\alpha^{n - 1}$ …(i)
$\Rightarrow \alpha S = \alpha + 3\alpha^{2} + 5\alpha^{3} + ... + (2n - 1)\alpha^{n}$ …(ii)
On subtracting Eq.(ii) from Eq. (i), we get
$(1 - \alpha)S = 1 + 2\alpha + 2\alpha^{2} + ... + 2\alpha^{n - 1} - (2n - 1)\alpha^{n}$
$= 2\left( 1 + \alpha + \alpha^{2} + ... + \alpha^{n - 1} \right) - 1 - (2n - 1)\alpha^{n}$
$= \ \frac{2(1 - \alpha^{n})}{1 - \alpha} - 2n = - 2n\ \ \ \ \ \ \ (\because a^{n} = 1)$
$\Rightarrow \ \ S = \frac{- 2n}{(1 - \alpha)}$