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
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the value of n so that `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.
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.
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:
−2/3, −6, −54, ...
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 progression 0.004, 0.02, 0.1, ... is 12.5?
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S1 (S2 + S3).
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.
If a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
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 the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
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
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 3 years.
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
For a G.P. a = 2, r = `-2/3`, find S6
For a G.P. if a = 2, r = 3, Sn = 242 find n
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:
`1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...`
Express the following recurring decimal as a rational number:
`2.bar(4)`
Express the following recurring decimal as a rational number:
`51.0bar(2)`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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 P2 R3 : S3 is equal to ______.
