Question

Let $P\left(n\right)=2^{3n}-7n-1$ then $P\left(n\right)$ is divisible by

• A. $63$
• B. $36$
• C. $49$
• D. $25$

Answer 1

The correct option is C $49$

$P\left(n\right)=2^{3n}-7n-1=-1-7n+{(1+7)}^n$

$\Rightarrow P\left(n\right)=-1-7n+(1+n_{C_1}7+n_{C_2}{7}^2+...+n_{C_n}{7}^n)=n_{C_2}{7}^2+...+n_{C_n}{7}^n$

$\Rightarrow P\left(n\right)=7^2(n_{C_2}+n_{C_3}7+...+n_{C_n}{7}^{n-2})$

Therefore, $P\left(n\right)$ is divisible by $49$.
Ans: 49