Question

If ${}^{2n+1}{P}_{n-1}:{}^{2n-1}{P}_n=3:5$, then the value of $n$ is :

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is B $4$

${}^{2n+1}{P}_{n-1}:{}^{2n-1}{P}_n=3:5\Rightarrow \frac{{}^{2n+1}{P}_{n-1}}{{}^{2n-1}{P}_n}=\frac{3}{5}\Rightarrow \frac{(2n+1)!}{(n+2)!}\times \frac{(n-1)!}{(2n-1)!}=\frac{3}{5}\Rightarrow \frac{(2n+1)\times 2n}{(n+2)(n+1)n}=\frac{3}{5}\Rightarrow 10(2n+1)=3(n+2)(n+1)\Rightarrow 3n^2-11n-4=0\Rightarrow (n-4)(3n+1)=0\Rightarrow n=4$
$\because (3n+1)\ne 0$ as $n$ can'tbe negative$)$