Question -
Answer -
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 toprove that P (k + 1) is true.
P (k): 52k – 1 is divisible by 24
: 52k – 1 = 24λ … (i)
We have to prove,
52k + 1 – 1 is divisible by 24
52(k + 1) – 1 = 24μ
So,
= 52(k + 1) – 1
= 52k.52 – 1
= 25.52k – 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.