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Last updated at May 29, 2018 by Teachoo

Ex 4.1, 9: Prove the following by using the principle of mathematical induction for all n β N: 1/2 + 1/4 + 1/8 + ....+ 1/2π = 1 β 1/2π Let P(n): 1/2 + 1/4 + 1/8 + ....+ 1/2π = 1 β 1/2π For n = 1, we have L.H.S = 1/2 R.H.S = 1 β 1/21 = 1/2 Hence, L.H.S. = R.H.S , β΄ P(n) is true for n = 1 Assume P(k) is true 1/2 + 1/4 + 1/8 + ....+ 1/2π = 1 β 1/2π We will prove that P(k + 1) is true. R.H.S = 1 β 1/2^(π + 1) L.H.S = 1/2 + 1/4 + 1/8 + ....+ 1/2^(π + 1) L.H.S = R.H.S β΄ P(k + 1) is true whenever P(k) is true. β΄ By the principle of mathematical induction, P(n) is true for n, where n is a natural number