Advertisements
Advertisements
Question
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Advertisements
Solution
We have,
\[ a_1 = a , a_2 = \frac{3 a^2}{4}, a_3 = \frac{9 a^3}{16}\]
\[\text { Now, } \frac{a_2}{a_1} = \frac{\frac{3 a^2}{4}}{a} = \frac{3a}{4}, \frac{a_3}{a_2} = \frac{\frac{9 a^3}{16}}{\frac{3 a^2}{4}} = \frac{3a}{4} \]
\[ \therefore \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{3a}{4}\]
\[\text { Thus, } a_1 , a_2 \text { and } a_3 \text { are in G . P . , where the first term is a and the common ratio is } \frac{3a}{4} .\]
APPEARS IN
RELATED QUESTIONS
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
If a, b, c are in A.P,; b, c, d are in G.P and ` 1/c, 1/d,1/e` are in A.P. prove that a, c, e are in G.P.
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
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, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.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.
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If logxa, ax/2 and logb x are in G.P., then write the value of x.
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.
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 second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
Mark the correct alternative in the following question:
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then p2R3 : S3 is equal to
For the G.P. if a = `2/3`, t6 = 162, find r.
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
For a G.P. a = 2, r = `-2/3`, find S6
If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [(1.08)5 = 1.47]
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is ______.
Let A1, A2, A3, .... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = `1/1296` and A2 + A4 = `7/36`, then the value of A6 + A8 + A10 is equal to ______.
