sol of 12^n+25^n-1 is divisible by 13 induction method

Hello Desingh dear, given 12^n + 25^n-1 is divisible by 13
Say n = 1, then 12 + 1 = 13. Yes divisible
Now n = 2, then 144 + 25 = 169. Correct divisible
So say n = k, now 12^k + 25^k-1 is diviible by 13
Then n = k+1, we get 12^k+1 + 25^k
==> 12^k * 12 + (12+13)^k
Or 12^k * 12 + 12^k * (1 +13/12)^k
Or 12^k ( 12 + 1 + k (13/12) + k(k-1)/1.2 * (13/12)^2 + ....)
Or 12^k (13 + k (13/12) + ...)
Or 13 * 12^k ( 1 + k(1/12) + .....)
It is so clear that it is divisible by 13.
Hence proved by induction
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