Answer :

Given; P(n) = 4^{n} – 1 is divisible by 3.

P(0) = 4^{0} – 1 = 0; is divisible by 3.

P(1) = 4^{1} – 1 = 3; is divisible by 3.

P(2) = 4^{2} – 1 = 15; is divisible by 3.

P(3) = 4^{3} – 1 = 63; is divisible by 3.

Let P(k) = 4^{k} – 1 is divisible by 3;

⇒ 4^{k} – 1 = 3x.

⇒ P(k+1) = 4^{k+1} – 1

= 4(3x + 1) – 1

= 12x + 3; is divisible by 3.

⇒ P(k+1) is true when P(k) is true

∴ By Mathematical Induction P(n) = 4^{n} – 1 is divisible by 3 is true for each natural number n.

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