Question

Prove that $5^{2n+1}-24n-25$ is divisible by $576$.

Answer 1

Let $5^{2n+2}-24n-25$ be denoted by $f\left(n\right)$;
then $f(n+1)=5^{2n+4}-24(n+1)-25$
$=5^2\cdot 5^{2n+2}-24n-49$;
$\therefore f(n+1)-25f\left(n\right)=25(24n-25)-24n-49$
$=576(n+1)$.
Therefore if $f\left(n\right)$ is divisible by $576$, so also is $f(n+1)$; but by trial we see that the theorem is true when $n=1$, therefore it is true when $n=2$, therefore it is true when $n=3$, and so on; thus it is true universally.
The above result may also be proved as follows:
$5^{2n+2}-24n-25={25}^{n+2}-24n-25$
$=25(1+24{)}^n-24n-25$
$=25+25.n.24+M\left({24}^2\right)-24n-25$
$=576n+M\left(576\right)$
$=M\left(576\right)$.