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Question
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
Solution
Let a and r be the 1st term and common ratio of the G.P. respectively.
∴ Sn = `"a"(("r"^"n" - 1)/("r" - 1)), "S"_(2"n") = "a"(("r"^(2"n") - 1)/("r" - 1)), "S"_(3"n") = "a"(("r"^(3"n") - 1)/("r" - 1))`
∴ S2n – Sn = `"a"(("r"^(2"n") - 1)/("r" - 1)) - "a"(("r"^"n" - 1)/("r" - 1))`
= `"a"/("r" - 1)("r"^(2"n") - 1 - "r"^"n" + 1)`
= `"a"/("r" - 1)("r"^(2"n") - "r"^"n")`
= `"ar"^"n"/("r" - 1) ("r"^"n" - 1)`
∴ S2n – Sn = `"r"^"n"*("a"("r"^"n" - 1))/("r" - 1)` ....(i)
S3n – S2n = `"a"(("r"^(3"n") - 1)/("r" - 1)) - "a"(("r"^(2"n") - 1)/("r" - 1))`
= `"a"/("r" - 1)("r"^(3"n") - 1 - "r"^(2"n") + 1)`
= `"a"/("r" - 1)("r"^(3"n") - "r"^(2"n"))`
= `"a"/("r" - 1)*"r"^(2"n")("r"^"n" - 1)`
= `"a"*(("r"^"n" - 1)/("r" - 1))*"r"^(2"n")`
∴ Sn(S3n – S2n) = `["a"*(("r"^"n" - 1)/("r" - 1))]["a"*(("r"^"n" - 1)/("r" - 1))"r"^(2"n")]`
= `["r"^"n"*("a"("r"^"n" - 1))/("r" - 1)]^2`
∴ Sn(S3n – S2n) = (S2n – Sn)2 ....[From (i)]
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