Let P(n) = 2^ 3n − 7n − 1 then P(n) is divisible by: - Vidyadip

MCQ

Let $P(n) = 2^{3n}− 7n − 1$ then $P(n)$ is divisible by:

A
$63$
B
$36$
✓
$49$
D
$25$

✓

Answer

Correct option: C.

$49$

$P(n) = 2^{3n}− 7n − 1 = −1 − 7n + (1 + 7)^n$
$\Rightarrow \ce{P(n)=-1-7 n + \left(1 + n C_1 7 + n C_2 7^2 + ... + n C_n 7^n\right)=n C_2 7^2 + ... + n C_n 7^n}$
$\Rightarrow \ce{P(n)=7^2\left({nC}_2 + {nC}_3 7 + ... + {nC}_{n} 7^{n}-2\right)}$
Therefore, $P(n)$ is divisible by $49.$

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