# 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. - Mathematics and Statistics

Sum

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)]

Concept: Sum of the First n Terms of a G.P.
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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board
Chapter 4 Sequences and Series
Exercise 4.2 | Q 10 | Page 55
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