Advertisements
Advertisements
प्रश्न
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
विकल्प
(a) 1/2
(b) 3/4
(c) 1
(d) none of these
Advertisements
उत्तर
(a) \[\frac{1}{2}\]
\[\text{ Let } S = \frac{1}{\left( 1 + x \right)} - \frac{\left( 1 - x \right)}{\left( 1 + x \right)^2} + \frac{\left( 1 - x \right)^2}{\left( 1 + x \right)^3} - \frac{\left( 1 - x \right)^3}{\left( 1 + x \right)^4} + . . . \infty \]
\[\text{ It is clear that it is a G . P . with a } = \frac{1}{\left( 1 + x \right)} \text{ and }r = - \frac{\left( 1 - x \right)}{\left( 1 + x \right)} . \]
\[ \therefore S = \frac{a}{\left( 1 - r \right)}\]
\[ \Rightarrow S = \frac{\frac{1}{\left( 1 + x \right)}}{\left[ 1 - \left( - \frac{\left( 1 - x \right)}{\left( 1 + x \right)} \right) \right]}\]
\[ \Rightarrow S = \frac{\frac{1}{\left( 1 + x \right)}}{\left[ 1 + \frac{\left( 1 - x \right)}{\left( 1 + x \right)} \right]}\]
\[ \Rightarrow S = \frac{\frac{1}{\left( 1 + x \right)}}{\left[ \frac{\left( 1 + x \right) + \left( 1 - x \right)}{\left( 1 + x \right)} \right]}\]
\[ \Rightarrow S = \frac{1}{2}\]
\[\]
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
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`.
If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
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, ...
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
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:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
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 series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
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, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
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 (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
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
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
Which term of the G.P. 5, 25, 125, 625, … is 510?
The numbers 3, x, and x + 6 form are in G.P. Find nth term
For a G.P. If t4 = 16, t9 = 512, find S10
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
Express the following recurring decimal as a rational number:
`0.bar(7)`
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
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:
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
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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 ______.
