Advertisements
Advertisements
प्रश्न
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
उत्तर
\[\text{ Here, a = 1, b, b^2 , b^3 , . . . \infty form an infinite G . P } . \]
\[ \]
\[ \therefore S_\infty = a = 1 + b + b^2 + b^3 + . . . \infty = \frac{1}{1 - b}\]
\[ \Rightarrow a = \frac{1}{1 - b}\]
\[ \Rightarrow 1 - b = \frac{1}{a} \]
\[ \Rightarrow b = 1 - \frac{1}{a}\]
\[ \therefore b = \frac{a - 1}{a}\]
APPEARS IN
संबंधित प्रश्न
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.
Given a G.P. with a = 729 and 7th term 64, determine S7.
Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
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`.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in 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, ...
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
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 seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Find three numbers in G.P. whose sum is 38 and their product is 1728.
Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
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:
\[\sum^{10}_{n = 2} 4^n\]
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
Three numbers are in A.P. and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P. Find the numbers.
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio \[(3 + 2\sqrt{2}) : (3 - 2\sqrt{2})\] .
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
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 a = 2, r = 3, Sn = 242 find n
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
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.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
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 an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
