Advertisements
Advertisements
Question
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.
Solution
Suppose the three terms of the geometric progression are `"a"/"r"`, a and ar.
Sum `"a"/"r" + "a" + "ar" = 39/10` ........(i)
And product = `"a"/"r" xx "a" xx "ar" = "a"^3 = 1`
or a = 1 ..........(ii)
By keeping a = 1 in equation (i)
`1/"r" + 1 + "r" = 39/10`
on multiplying by 10r
= 10 + 10r + 10r2 = 39r
= 10r2 − 29r + 10 =
= 10r2 - 25r - 4r + 10 = 0
= 5r (2r - 5) -2 (2r- 5) = 0
= (5r - 2) (2r - 5) = 0
r = `5/2` or `2/5`
a = 1
`1/"r" = 5/2, "r" = 2/5`
∴ Terms of geometric progression = `5/2, 1, 2/5` or `2/5, 1, 5/2`
APPEARS IN
RELATED QUESTIONS
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Evaluate `sum_(k=1)^11 (2+3^k )`
Given a G.P. with a = 729 and 7th term 64, determine S7.
If a and b are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, c, d, form a G.P. Prove that (q + p): (q – p) = 17 : 15.
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
The ratio of the sum of first three terms is to that of first 6 terms of a G.P. is 125 : 152. Find the common ratio.
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 \[\frac{1}{r^n}\].
Find the rational number whose decimal expansion is \[0 . 423\].
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 three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.
If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are in G.P.
If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
Write the product of n geometric means between two numbers a and b.
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
The numbers 3, x, and x + 6 form are in G.P. Find x
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 10 years.
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Find : `sum_("n" = 1)^oo 0.4^"n"`
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Select the correct answer from the given alternative.
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is –
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
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
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
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
