$\forall n \in N; 5^{2 n + 2} - 24 n - 25$ is divisible by

576

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25

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24

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50

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Solution

The correct options are
B $24$
C $576$
$a = 5^{2 n + 2} - 24 n - 25 = - 25 - 24 n + {(1 + 24)}^{n + 1}$

$\Rightarrow a = - 25 - 24 n + (1 + {n + 1}_{C_{1}} 24 + n + 1_{C_{2}} 24^{2} + . . . + n + 1_{C_{n + 1}} 24^{n + 1})$

$= - 25 - 24 n + 1 + (n + 1) 24 + {n + 1}_{C_{2}} 24^{2} + . . . + {n + 1}_{C_{n + 1}} 24^{n + 1}$

$\Rightarrow a = 24^{2} (n_{C_{2}} + n_{C_{3}} 24 + . . . + n_{C_{n}} 24^{n - 1})$

Therefore, $a$ is divisible by 576 & 24.

Ans: A,C

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