P(n):52n+1+3n+2.2n−1 is divisible by

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Question

$P (n) : 5^{2 n + 1} + 3^{n + 2} {.2}^{n - 1}$ is divisible by

Options:

A . 19

B . 18

C . 17

D . 14

Answer: Option A
:
A

$P (n) : 5^{2 n + 1} + 3^{n + 2} {.2}^{n - 1}$
$P (1) : 5^{3} + 3^{3} .1 = 152 \to divisible by 19$
Assume P(k) is true.
$5^{2 k + 1} + 3^{k + 2} {.2}^{k - 1} is divisible by 19 5^{2 k + 1} + 3^{k + 2} {.2}^{k - 1} = 19 m P (k + 1) : 5^{2 k + 3} + 3^{k + 3} {.2}^{k} = {25.5}^{2 k + 1} + {6.3}^{k + 2} {.2}^{k - 1} = 6. (5^{2 k + 1} + 3^{k + 2} {.2}^{k - 1}) + {19.5}^{2 k + 1} = 19 (6 m + 5^{2 k + 1})$

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