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Ex 4.1

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Last updated at March 22, 2023 by Teachoo

Ex 4.1, 12: Prove the following by using the principle of mathematical induction for all n β N: a + ar + ar2 + β¦β¦..+ arn β 1 = (π(π^π β 1))/(π β 1) Let P (n) : a + ar + ar2 + β¦β¦..+ arn β 1 = π(π^π β 1)/(π β 1) For n = 1, L.H.S = a R.H.S = (π(π1 β 1))/(π β 1) = (π(π β 1))/(π β 1) = a L.H.S. = R.H.S β΄ P(n) is true for n = 1 Assume that P(k) is true a + ar + ar2 + β¦β¦..+ ark β 1 = π(π^π β 1)/(π β 1) We will prove that P(k + 1) is true. a + ar + ar2 + β¦β¦..+ ar(k + 1) β 1 = π(π^(π + 1) β 1)/(π β 1) a + ar + ar2 + β¦β¦..+ ark β 1 + ark = π(π^(π + 1) β 1)/(π β 1) We have to prove P(k+1) from P(k) i.e. (2) from (1) From (1) a + ar + ar2 + β¦β¦..+ ark β 1 = π(π^π β 1)/(π β 1) Adding ark both sides a + ar + ar2 + β¦β¦.. +ark β 1 + ark = π(π^π β 1)/(π β 1) + ark = (π(π^π β 1) + (π β 1)ππ^π)/(π β 1) = (ππ^π β π + ππ^π (π) β ππ^π)/(π β 1) = (ππ^πβ ππ^π β π + ππ^π (π))/(π β 1) = (0 β π + ππ^π (π))/(π β 1) = (β π + ππ^π (π))/(π β 1) = (β π + ππ^π (π^1 ))/(π β 1) = (β π + ππ^(π + 1))/(π β 1) = (π (β1 + π^(π + 1) ))/(π β 1) = π(π^(π + 1) β 1)/(π β 1) Thus, a + ar + ar2 + β¦β¦..+ ark β 1 + ark = π(π^(π + 1) β 1)/(π β 1) which is the same as P(k + 1) 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