Let P(n): 3^{2n + 2} – 8n – 9

For n = 1

= 3^{2.1 + 2} - 8.1 - 9

= 81 – 17

= 64

Since, it is divisible by 8

Let P(n) is true for n=k, so

= 3^{2k + 2} – 8k – 9 is divisible by 8

= 3^{2k + 2} – 8k – 9 = 8λ - - - - - (1)

We have to show that,

= 3^{2k + 4} – 8(k + 1) – 9 is divisible by 8

= 3^{(2k + 2) + 2} – 8(k + 1) – 9 = 8μ

Now,

= 3^{2(k + 1)}.3² – 8(k + 1) – 9

= (8λ + 8k + 9)9 – 8k – 8 – 9

= 72λ + 72k + 81 - 8k - 17 using equation (1)

= 72λ + 64k + 64

= 8(9λ + 8k + 8)

= 8μ

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

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