Advertisements
Advertisements
प्रश्न
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
Advertisements
उत्तर
\[0 . \overline {231 }\]
\[\text { Let } S = 0 . \overline {231 }\]
\[ \Rightarrow S = 0 . 231 + 0 . 000231 + 0 . 000000231 + . . . \infty \]
\[ \Rightarrow S = 0 . 231\left( 1 + {10}^{- 3} + {10}^{- 6} + . . . \infty \right)\]
\[\text { It is a G . P } . \]
\[ \therefore S = 0 . 231\left( \frac{1}{1 - {10}^{- 3}} \right)\]
\[ \Rightarrow S = \frac{231}{999}\]
APPEARS IN
संबंधित प्रश्न
Which term of the following sequence:
`sqrt3, 3, 3sqrt3`, .... is 729?
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.
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.
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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
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. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
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 progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 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 geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Find the sum of the following series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
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.
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term is ______.
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
If for a sequence, tn = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
For the following G.P.s, find Sn
3, 6, 12, 24, ...
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Express the following recurring decimal as a rational number:
`0.bar(7)`
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
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
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 ______.
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 ______.
