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 ∈ 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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