3²ⁿ – 1 is divisible by 8, for all natural numbers n.

Given; P(n) = 3²ⁿ – 1 is divisible by 8.

P(0) = 3⁰ – 1 = 0; is divisible by 8.

P(1) = 3² – 1 = 8; is divisible by 8.

P(2) = 3⁴ – 1 = 80; is divisible by 8.

P(3) = 3⁶ – 1 = 728; is divisible by 8.

Let P(k) = 3^2k – 1 is divisible by 8; ⇒ 3^2k – 1 = 8x.

⇒ P(k+1) = 3^2(k+1) – 1 = 3²(8x + 1) – 1 = 72x + 8; is divisible by 8.

⇒ P(k+1) is true when P(k) is true.

∴ By Mathematical Induction P(n) = 3²ⁿ – 1 is divisible by 8, for all natural numbers n.