Question:

52n −1 is divisible by 24 for all n ∈ N.

Solution:

Let P(n) be the given statement.

Now,

$P(n): 5^{2 n}-1$ is divisible by 24 for all $n \in N$.

Step 1:

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

It is divisible by 24 .

Thus, $P(1)$ is true.

Step 2 :

Let $P(m)$ be true.

Then, $5^{2 m}-1$ is divisible by 24 .

Now, let $5^{2 m}-1=24 \lambda$, where $\lambda \in N$.

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

Now,

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

$=5^{2 m} 5^{2}-1$

$=25(24 \lambda+1)-1$

$=600 \lambda+24$

$=24(25 \lambda+1)$

It is divisible by 24 .

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

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