Question

3²ⁿ – 1 is divisible by 8

Answer 1

P(n) = 3²ⁿ – 1 is divisible by 8 

For n = 1, we get 

P(1) = 3^2.1 – 1 = 9 – 1 = 8 

P(1) = 8, which is divisible by 8. 

Let P(n) be true for n = k 

P(k) = 3^2k – 1 is divisible by 8 ... (1) 

Now, P(k + 1) = 3^{(2k + 2)} – 1 = 3^2k.3² – 1 

= 3² (3^2k – 1) + 8

Now, 3^2k – 1 is divisible by 9. [Using (1)] 

∴ 3² (3^2k – 1) + 8 is also divisible by 8. 

Hence, 3²ⁿ – 1 is divisible by 8 ∀ n E N