Solve the following

Solution:

Let P(n) be the given statement.

Now,

$P(n): 5^{2 n+2}-24 n-25$ is divisible by 576 for all $n \in N$.

Step 1:

$P(1)=5^{2+2}-24-25=625-49=576$

It is divisible by 576 .

Thus, $P(1)$ is true.

Step 2 :

Let $P(m)$ be true.

Then,

$5^{2 m+2}-24 m-25$ is divisible by 576

Let $5^{2 m+2}-24 m-25=576 \lambda$, where $\lambda \in N$.

We need to show that $P(m+1)$ is true wheneve $r P(m)$ is true.

Now,

$P(m+1)=5^{2 m+4}-24(m+1)-25$

$=5^{2} \times(576 \lambda+24 m+25)-24 m-49$

$=25 \times 576 \lambda+600 m+625-24 m-49$

$=25 \times 576 \lambda+576 m+576$

$=576(25 \lambda+m+1)$

It is divisible by 576 .

Thus, $P(m+1)$ is true

By the principle of $m$ athematical $i$ nduction, $P(n)$ is true for all $n \in N$.