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 sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
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.
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
Find the geometric means of the following pairs of number:
−8 and −2
If logxa, ax/2 and logb x are in G.P., then write the value of x.
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio 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
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
The fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
For a G.P. if S5 = 1023 , r = 4, Find a
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Find : `sum_("n" = 1)^oo 0.4^"n"`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Answer the following:
Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
The sum or difference of two G.P.s, is again a G.P.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
