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))`, S2n = `"a"(("r"^(2"n") - 1)/("r" - 1))`, S3n = `"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
Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.
Find the value of n so that `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
Find three numbers in G.P. whose sum is 38 and their product is 1728.
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
If S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Find the sum of the following serie to infinity:
\[\frac{1}{3} + \frac{1}{5^2} + \frac{1}{3^3} + \frac{1}{5^4} + \frac{1}{3^5} + \frac{1}{56} + . . . \infty\]
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.
The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after n years.
For a G.P. If t4 = 16, t9 = 512, find S10
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Express the following recurring decimal as a rational number:
`2.bar(4)`
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
