Advertisements
Advertisements
प्रश्न
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.
उत्तर
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)]
APPEARS IN
संबंधित प्रश्न
For a G.P., if a = 2, r = `-2/3`, find S6.
For a G.P., if the sum of the first 3 terms is 125 and the sum of the next 3 terms is 27, find the value of r.
For a G.P., if t3 = 20, t6 = 160, find S7.
If for a sequence `t_n = 5^(n-3) / 2^(n-3),` show that the sequence is a G.P.
Find its first term and the common ratio.
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.
If for a sequence, tn = `(5^(n-3))/(2^(n-3))`, show that the sequence is a G.P. Find its first term and the common ratio.
If for a sequence, `t_n=(5^n-3)/(2^n-3)` show that the sequence is a G.P.
Find its first term and the common ratio.
If Sn, S2n, S3n are the sum of n, 2n, and 3n terms of a G.P. respectively, then verify that `S_n (S_(3n) - S_(2n)) = (S_(2n) - S_n)^2`.
If for a sequence, tn = `5^(n-3)/2^(n-3)`, show that the sequence is a G.P.
Find its first term and the common ratio.
If for a sequence, `t_n = 5^(n-3)/2^(n-3)`, show that the sequence is a G.P. Find its first term and the common ratio.
If for a sequence, `t_n=5^(n-3)/2^(n-3)`, show that the sequence is a G.P.
Find its first term and the common ratio.
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.
If for a sequence, `t_n = 5^(n-3)/2^(n-3)`, show that the sequence is a G.P. Find its first term and the common ratio.
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
If `S_n, S_(2n), S_(3n)` are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that `S_n (S_(3n) - S_(2n)) = (S_(2n) - S_n)^2`.
If `S_n, S_(2n), S_(3n)` are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that `S_n(S_(3n) - S_(2n)) = (S_(2n) - S_n)^2`.
