Explanation

(c)
LHS $= \ \frac{1(1 - \lambda^{n + 1})}{1 - \lambda} = \frac{1 - \lambda^{n + 1}}{1 - \lambda}$
And RHS $= (1 + \lambda)\left( 1 + \lambda^{2} \right)\left( 1 + \lambda^{4} \right)$
$\left( 1 + \lambda^{8} \right)(1 + \lambda^{16})$
$= \ \frac{\begin{array}{r}
(1 - \lambda)(1 + \lambda)\left( 1 + \lambda^{2} \right)\ \\
\left( 1 + \lambda^{4} \right)\left( 1 + \lambda^{8} \right)(1 + \lambda^{16})
\end{array}}{(1 - \lambda)}$
$= \frac{\begin{array}{r}
\left( 1 - \lambda^{2} \right)\left( 1 + \lambda^{2} \right)\left( 1 + \lambda^{4} \right) \\
\left( 1 + \lambda^{8} \right)(1 + \lambda^{16})
\end{array}}{1 - \lambda}$
$= \ \frac{(1 - \lambda^{32})}{1 - \lambda}$
$\therefore\ \ \frac{1 - \lambda^{n + 1}}{1 - \lambda} = \frac{1 - \lambda^{32}}{1 - \lambda}$
$\Rightarrow \ 1 - \lambda^{n + 1} = 1 - \lambda^{32}$
$\therefore\ \ n + 1 = 32\ \ \Rightarrow n = 31$