Let P (n): 3^{2n} + 7 is divisible by 8

Let us check for n = 1,

P (1): 3^{2} + 7 = 9 + 7 = 16

P (n) is true for n = 1. where, P (n) is divisible by 8

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

P (k): 3^{2k} + 7 is divisible by 8

: 3^{2k} + 7 = 8λ

: 3^{2k} = 8λ – 7 … (i)

We have to prove,

3^{2(k + 1)} + 7 is divisible by 8

3^{2k + 2} + 7 = 8μ

So,

= 3^{2(k + 1)} + 7

= 3^{2k}.3^{2} + 7

= 9.3^{2k} + 7

= 9.(8λ – 7) + 7 by using equation (i)

= 72λ – 63 + 7

= 72λ – 56

= 8(9λ – 7)

= 8μ

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

Hence, P (n) is true for all n ∈ N.

Answered by Aaryan | 1 year agoGiven an example of a statement P (n) such that it is true for all n ϵ N.

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