Advertisements
Advertisements
Question
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn
Advertisements
Solution
Let the geometric progression be a + ar + ar2 +….. + arn – 1
The product of these n terms, P = a. ar . ar2….. arn – 1
= `"a"^"n". "r"^(1 + 2 + ...... + ("n" - 1))`
= `"a"^"n""r" ("n" ("n" - 1))/2`
∴ `"P"^2 = "a"^(2"n"). "r"("n"("n" - 1))`
R = `1/"a" + 1/"ar" + 1/"ar"^2 + ....... + 1/"ar"^("n" - 1)`
= `(1/"a" [(1/"r")^"n" - 1])/1/"r" -1`
= `((1 - "r"^"n")"r")/("ar"^"n"(1 - "r"))`
∴ Rn = `((1 - "r"^"n")^"n")/(("a"^"n" "r"^"n"("n" - 1))(1 - "r")^"n")`
Left Side: P2 Rn = `"a"^(2"n") "r"^("n" ("n" -1)) ((1 - "r"^"n")^"n")/(("a"^"n""r"^("n"("n" - 1))(1 - "r")^"n"))`
= `("a"^"n"(1 - "r"^"n")^"n")/((1 - "r")"n") = "S"^"n"`
Whereas S = a + ar + ar2 + .... + arn - 1
= `("a"(1 - "r"^"n"))/(1 - "r")`
Hence, P2Rn = Sn
APPEARS IN
RELATED QUESTIONS
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
The sum of first three terms of a G.P. is `39/10` and their product is 1. Find the common ratio and the terms.
Given a G.P. with a = 729 and 7th term 64, determine S7.
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is `1/r^n`.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following series:
7 + 77 + 777 + ... to n terms;
The ratio of the sum of the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.
Find the sum :
\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]
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 rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
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 \[\frac{32}{9}\text{and}\frac{81}{2}\] .
Find the geometric means of the following pairs of number:
2 and 8
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
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 3 years.
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the perimeters of all the squares
Select the correct answer from the given alternative.
If common ratio of the G.P is 5, 5th term is 1875, the first term is -
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
The third term of G.P. is 4. The product of its first 5 terms is ______.
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
