Let P(n): 3²ⁿ + 7 is divisible by 8

Let’s check For n =1

P(1): 3² + 7 = 9 + 7

= 16

Since, it is divisible by 8

So, P(n) is true for n=1

Now, for n=k

P(k): 3^2k + 7 = 8λ - - - - - - - (1)

We have to show that,

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

3^{2k + 2} + 7 = 8μ

Now,

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

= 3^2k.3² + 7

= 9.3^2k + 7

= 9.(8λ - 7) + 7

= 72λ - 63 + 7

= 72λ - 56

= 8(9λ - 7)

= 8μ

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

Hence, P(n) is true for all n∈N by PMI