Question

Prove that 3²ⁿ⁺² – 8n – 9 is divisible by 64 for any positive integer n.

Answer 1

Let T(n) be the statement: 3²ⁿ⁺² – 8n – 9 is divisible by 64. 

Basic Step: 

For n =1, 3^2×1+2 – 8 × 1 – 9 = 81 – 17 = 64 which is divisible by 64. 

⇒ T(1) holds. 

Induction Step: 

Let T(k), k∈N hold, i.e., 

3^2k+2 – 8k – 9 is divisible by 64. 

Then, T(k + 1) = 3^{2(k+1) + 2} – 8(k + 1) – 9 = 3². 3^2k+2 – 8k – 17 

= 9 (3^2k+2 – 8k – 9) + 64k + 64 = 9. 

T (k) + 64 (k + 1) 

⇒ T(k + 1) is divisible by 64, whenever T(k) is divisible by 64. 

⇒ T(n) is true for every natural number n.