Thank you for reporting, we will resolve it shortly
Q.
If n is a positive integer, then $5^{2n + 2} - 24n - 25$ is divisible by
Principle of Mathematical Induction
Solution:
Let $P(n)$ be the statement given by
$P(n) : 5^{2n + 2} - 24n - 25$ is divisible by $576.$
For $n = 1,$
$P(1) : 5^{2 + 2} - 24 - 25 = 625 - 49 = 576,$ which is divisible by $576.$
$\therefore P(1)$ is true.
Let $P(k)$ be true,
i.e. $P(k) : 5^{2k + 2} - 24k - 25$ is divisible by $576.$
$\Rightarrow 5^{2k+2}-24k-25=576\lambda\,...\left(i\right)$
We have to show that $P\left(k + 1\right)$ is true,
i.e. $5^{2k+4}-24k-49$ is divisible by $576$
Now, $5^{2k+4}-24k-49$
$=\left(576\lambda+24k+25\right)\cdot25-24k-49$ [from (i)] $=576.25\lambda+600k+625-24k-49$
$=576\left\{25\lambda+k+1\right\}$, which is divisible by $576$.
$\therefore P\left(k+1\right)$ is true whenever $P\left(k\right)$ is true.
So, $P\left(n\right)$ is true for all $n \in N.$