Q. 3

# 4n – 1

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

P(0) = 40 – 1 = 0; is divisible by 3.

P(1) = 41 – 1 = 3; is divisible by 3.

P(2) = 42 – 1 = 15; is divisible by 3.

P(3) = 43 – 1 = 63; is divisible by 3.

Let P(k) = 4k – 1 is divisible by 3;

4k – 1 = 3x.

P(k+1) = 4k+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) = 4n – 1 is divisible by 3 is true for each natural number n.

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