Advertisements
Advertisements
प्रश्न
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
Advertisements
उत्तर
\[0 . 6\overline8\]
\[\text { Let } S = 0 . 6\overline8\]
\[ \Rightarrow S = 0 . 6 + 0 . 08 + 0 . 008 + 0 . 0008 + 0 . 00008 + . . . \infty \]
\[ \Rightarrow S = 0 . 6 + 0 . 08\left( 1 + {10}^{- 1} + {10}^{- 2} + {10}^{- 3} + . . . \infty \right)\]
\[\text { It is a G . P } . \]
\[ \therefore S = 0 . 6 + 0 . 08\left( \frac{1}{1 - {10}^{- 1}} \right)\]
\[ \Rightarrow S = 0 . 6 + \frac{0 . 8}{9}\]
\[ \Rightarrow S = \frac{6 . 2}{9}\]
\[ \Rightarrow S = \frac{62}{90} = \frac{31}{45}\]
संबंधित प्रश्न
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
Find three numbers in G.P. whose sum is 38 and their product is 1728.
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, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
Find the sum of the following series:
7 + 77 + 777 + ... 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).
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
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:
10 − 9 + 8.1 − 7.29 + ... ∞
Express the recurring decimal 0.125125125 ... as a rational number.
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If a, b, c, d are in G.P., prove that:
(a2 + b2), (b2 + c2), (c2 + d2) are in G.P.
If a, b, c are in G.P., then prove that:
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
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 ______.
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.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
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 the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
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:
`2.3bar(5)`
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"`
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Select the correct answer from the given alternative.
Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively. (a, b > 0)
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
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 G.P. a = `4/3` and t7 = `243/1024`, find the value of r
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 ______.
