2.4^ 2n+1 +3^ 3n+1 is divisible by: (for all n N) - Vidyadip

MCQ

$2.4^{2\text{n+1}}+3^{3\text{n+1}}$ is divisible by: $($for all $n \in N)$

A
$2$
B
$9$
C
$3$
✓
$11$

✓

Answer

Correct option: D.

$11$

Concepts:
Suppose there is a given statement $p(n)$ involving the natural number $n$ such that
The statement is true for $n = 1,$
i.e., $P(1)$ is true, and
If the statement is true for $n = k ($where $k$ is some positive integer$)$, then the statement is also true for $n = k + 1,$
i.e., truth of $P (k)$ implies the truth of $P(k + 1).$
Then, $P (n)$ is true for all natural numbers $n.$
Calculation:
Given:
$\text{p}\text{(n)}=2.4^{2\text{n+1}}+3^{3\text{n+1}}$
Take $n = 1$
$\text{p}(1)=2.4^{2\times+1}+3^{3\times+1}$
$=2.4^3+3^4=209=11\times19$
Therefore we can say that $P (n)$ is divisible by $11.$

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