Question

The number of positive terms in the sequence $\left\{X_n\right\},$
if $X_n=\frac{90}{7{(}^n{P}_n)}-\frac{{}^{n+4}{P}_3}{{}^{n+2}{P}_{n+2}},n\in N$ is

• A. $6$
• B. $5$
• C. $7$
• D. $4$

Answer 1

The correct option is B $5$

$X_n=\frac{90}{7{(}^n{P}_n)}-\frac{{}^{n+4}{P}_3}{{}^{n+2}{P}_{n+2}}\Rightarrow X_n=\frac{90}{7(n!)}-\frac{(n+4)(n+3)(n+2)}{(n+2)!}\Rightarrow X_n=\frac{90(n+1)}{7(n+1)!}-\frac{(n+4)(n+3)}{(n+1)!}\Rightarrow X_n=\frac{90n+90-7(n^2+7n+12)}{7(n+1)!}\Rightarrow X_n=\frac{41n+6-7n^2}{7(n+1)!}$

For positive terms,
$41n+6-7n^2>0\Rightarrow 7n^2-41n-6<0\Rightarrow (n-6)(7n+1)<0\Rightarrow 1\le n<6(\because n\in N)$

So, for $n<6$ the series will have positive terms.
Hence, the number of positive term will be $5.$