For each natural number, the statement P(n) = 2^ 3n − 1 is divisi… - Vidyadip

MCQ

For each natural number, the statement $P(n) = 2^{3n}− 1$ is divisible by:

A
$10$
B
$6$
✓
$7$
D
None of these.

✓

Answer

Correct option: C.

$7$

$ P(n)=2^{3 n}-1=8^n-1 $
$ P(n)=(1+7)^n-1 $
$ \Rightarrow P(n)=1+{ }^n C_1 \times 7+{ }^n C_2 \times 7^2+\ldots+{ }^n C_n \times 7^n-1 $
$ \Rightarrow P(n)=7\left({ }^n C_1+{ }^n C_2 7+\ldots+{ }^n C_n 7{ }^{n-1}\right)$
Therefore, $P(n)$ is divisible by $7.$

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