Explanation

(a)
Given, $\ ^{2n + 1}P_{n - 1}:\ ^{2n - 1}P_{n} = 3:5$
$\Rightarrow \frac{(2n + 1)!}{(n + 2)!} \times \frac{(n - 1)!}{(2n - 1)!} = \frac{3}{5}$
$\Rightarrow \frac{(2n + 1)2n}{(n + 2)(n + 1)n} = \frac{3}{5}$
$\Rightarrow 10\ (2n + 1) = 3(n^{2} + 3n + 2)$
$\Rightarrow 3n^{2} - 11n - 4 = 0$
$\Rightarrow (3n + 1)(n - 4) = 0$
$\Rightarrow \ \ n = 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( n \neq - \frac{1}{3} \right)$