Question

Prove by the principle of mathematical induction that for every natural number n, 3^{2n + 2} – 8n – 9 is divisible by 8.

Answer 1

Let P(n): 3^{2n + 2} – 8n – 9 is divisible by 8.

Then, P(1): 3^{2.1 + 2} – 8.1 – 9 is divisible by 8.

(i.e.) 3⁴ – 8 – 9 is divisible by 8 or 81 – 8 – 9 is divisible by 8

(or) 64 is divisible by 8, which is true. 

Suppose P(k) is true, then 

P(k) : 3^{2k + 2} – 8k – 9 is divisible by 8 

(i.e.) 3^{2k + 2} – 8k – 9 = 8m, where m ∈ N (or) 

3^{2k + 2} = 8m + 8k + 9 

P(k + 1) is the statement given by, … (1) 

P(k + 1) : 3^{2(k + 1) + 2} – 8(k + 1) – 9

Now,

∴ P(k + 1) is true 

Hence, by the principle of mathematical induction, P(n) is true for all n ∈ N