52n – 1 is divisible by 24 for all n ϵ N

Let P (n): 5²ⁿ – 1 is divisible by 24

Let us check for n = 1,

P (1): 5² – 1 = 25 – 1 = 24

P (n) is true for n = 1. Where, P (n) is divisible by 24

Now, let us check for P (n) is true for n = k, and have to prove that P (k + 1) is true.

P (k): 5^2k – 1 is divisible by 24

: 5^2k – 1 = 24λ … (i)

We have to prove,

5^{2k + 1} – 1 is divisible by 24

5^{2(k + 1)} – 1 = 24μ

So,

= 5^{2(k + 1)} – 1

= 5^2k.5² – 1

= 25.5^2k – 1

= 25.(24λ + 1) – 1 by using equation (1)

= 25.24λ + 24

= 24λ

P (n) is true for n = k + 1

Hence, P (n) is true for all n ∈ N.