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Q. If $ n $ is a positive integer, then $ {{5}^{2n+2}}-24n-25 $ is divisible by

KEAMKEAM 2009Binomial Theorem

Solution:

Let $ P(n)={{5}^{2n+2}}-24n-25 $
For $ n=1, $ $ p(1)={{5}^{2}}-24-25=576 $ $ p(2)={{5}^{6}}-24(2)-25=15552 $
$=576\times 27 $
Here, we see that $ p(n) $ is divisible by 576.