If S, P, R are the sum, product and sum of the reciprocals of n terms of a G.P. respectively, then verify that (SR)n=P2. - Mathematics and Statistics

Sum

If S, P, R are the sum, product and sum of the reciprocals of n terms of a G.P. respectively, then verify that ("S"/"R")^"n" = "P"^2.

Solution

Let a be the 1st term and r be the common ratio of the G.P.
∴ the G.P. is a, ar, ar2, ar3, ..., arn–1

∴ S = a + ar + ar2 + ... + arn–1 = "a"(("r"^"n" - 1)/("r" - 1))

P = a(ar) (ar)2 ... (arn–1)

= "a"^"n"*"r"^(1 + 2 + 3 + ... + ("n" - 1))

= "a"^"n"*"r"^(("n"("n" - 1))/2)

∴ P = "a"^(2"n")*"r"^("n"("n" - 1)           ...(i)

R =  1/"a" + 1/"ar" + 1/"ar"^2 + ... + 1/"ar"^("n" - 1)

= ("r"^("n" - 1) + "r"^("n" - 2) + "r"^("n" - 3) + ... + "r"^2  + "r" + 1)/("a"*"r"^("n" - 1)

= (1 + "r" + "r"^2 + ... + "r"^("n" - 2) + "r"^("n" - 1))/("a"*"r"^("n" - 1)

1, r, r2, ..., rn–1 are in G.P., with a = 1, r = r

∴ R = 1/"ar"^("n" - 1)(("r"^"n" - 1)/("r" - 1)) = 1/("a"^2*"r"^("n" - 1)) xx "a" xx (("r"^"n" - 1)/("r" - 1))

∴ R = 1/("a"^2*"r"^("n" - 1))"S"

∴ "a"^2*"r"^("n" - 1) = "S"/"R"

∴ ("a"^2*"r"^("n" - 1))^"n" = ("S"/"R")^"n"

∴ "a"^(2"n")*"r"^("n"("n" - 1)) = ("S"/"R")^"n"

∴ P2 = ("S"/"R")^"n".          ...[From (i)]

Concept: Sum of the First n Terms of a G.P.
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Balbharati Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board
Chapter 4 Sequences and Series
Exercise 4.2 | Q 9 | Page 55