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

Asked by Sakshi | 1 year ago |  60

##### Solution :-

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): 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.

Answered by Aaryan | 1 year ago

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