Advertisements
Advertisements
Question
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Advertisements
Solution
Let the first term of the geometric progression = a
Common and ratio = r
5th term = ar5–1 = ar4 = p
8th term = ar8–1 = ar7 = q
11th term = ar11–1= ar10 = s
Left side = q2 = (ar7)2
= a2 × r14
Right side = ps = ar4 ar10
= a2 × r14
Hence, q2 = ps
APPEARS IN
RELATED QUESTIONS
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n 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.
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`.
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.
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
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
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 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
Find the rational number whose decimal expansion is `0.4bar23`.
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
If a, b, c are in G.P., prove that the following is also in G.P.:
a3, b3, c3
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are 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 \[a^{b - c} b^{c - a} c^{a - b} = 1\]
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
For a G.P. if a = 2, r = 3, Sn = 242 find n
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
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.
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
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 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 ______.
