Advertisements
Advertisements
Question
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
Advertisements
Solution
Let a be the first term and r be the common, ratio of the G.P.
Then S = `("a"("r"^"n" - 1))/("r" - 1)`
P = a × ar × ar2 × ...... arn–1
= `"a"^"n"*"r"^(1 + 2 + 3 + ... + ("n" - 1))`, where
1 + 2 + 3 + ... + (n – 1) = `(("n" - 1))/2[2(1) + ("n" - 1 - 1)1]`
= `(("n" - 1))/2(2 + "n" - 2) = ("n"("n" - 1))/2`
∴ P = `"a"^"n"*"r"("n"("n" - 1))/2`
and R = `1/"a" + 1/"ar" + 1/"ar"^2 + ... + 1/("ar"^("n" - 1))`
= `("r"^("n" - 1) + "r"^("n" - 2) + "r"^("n" - 3) + ... + 1)/"ar"^("n" - 1)`, where
rn–1 + rn–2 + rn–3 + ... + 1 = 1 + r + r2 + ... + rn–1
= `(1("r"^"n" - 1))/("r" - 1)`
= `("r"^"n" - 1)/("r" - 1)`
∴ R = `("r"^"n" - 1)/(("r" - 1)*"ar"^"n" - 1)`
∴ `("S"/"R")^"n" = [("a"("r"^"n" - 1))/("r" - 1) xx (("r" - 1)"ar"^("n" - 1))/("r"^"n" - 1)]^"n"`
= `("a"^2"r"^("n" - 1))^"n" = "a"^(2"n")*"r"^("n"("n" - 1))`
= `["a"^"n"*"r" ("n"("n" - 1))/2]^2` = P2
Hence, `["S"/"R"]^"n"` = P2
APPEARS IN
RELATED QUESTIONS
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Which term of the following sequence:
`sqrt3, 3, 3sqrt3`, .... is 729?
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
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 an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
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 which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.
If a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
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 the G.P. if r = − 3 and t6 = 1701, find a.
The numbers 3, x, and x + 6 form are in G.P. Find x
For a G.P. if a = 2, r = 3, Sn = 242 find n
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:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
Answer the following:
If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is ______.
The third term of a G.P. is 4, the product of the first five terms is ______.
The sum of infinite number of terms of a decreasing G.P. is 4 and the sum of the terms to m squares of its terms to infinity is `16/3`, then the G.P. is ______.
