Q4 of 30 Page 70

23n – 1 is divisible by7, for all natural numbers n.

Given; P(n) = 23n – 1 is divisible by 7.

P(0) = 20 – 1 = 0; is divisible by 7.


P(1) = 23 – 1 = 7; is divisible by 7.


P(2) = 26 – 1 = 63; is divisible by 7.


P(3) = 29 – 1 = 512; is divisible by 7.


Let P(k) = 23k – 1 is divisible by 7;


⇒ 23k – 1 = 7x.


⇒ P(k+1) = 23(k+1) – 1


= 23(7x + 1) – 1


= 56x + 7


= 7(8x + 1) ; is divisible by 7.


⇒ P(k+1) is true when P(k) is true.


∴ By Mathematical Induction P(n) = 23n – 1 is divisible by7, for all natural numbers n.


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