Question

The number of positive terms in the sequence $x_n=\frac{195}{{4.}^n{P}_n}-\frac{{}^{n+3}{P}_3}{{}^{n+1}{P}_{n+1}},n\in N$ is

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

Answer: (C)

$4$

We have:
$x_n=\frac{195}{4.n!}-\frac{(n+3)(n+2)(n+1)}{(n+1)!}=\frac{195}{4.n!}-\frac{(n+3)(n+2)}{n!}=\frac{195-4n^2-20n-24}{4.n!}=\frac{171-4n^2-20n}{4.n!}$
The denominator is always positive, hence can be ignored.
Since we need to make $x_n$ positive, we need to have: $171-4n^2-20n>0=>4n^2+20n-171<0$
$\Rightarrow (2n-9)(2n+19)<0\Rightarrow (n-4.5)(n+9.5)<0\Rightarrow -9.5<n<4.5$
Since $n$ must be a positive integer, there are $4$ values of $n$: $1,2,3,4$