Advertisements
Advertisements
Question
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
Advertisements
Solution
\[0 . \overline3\]
\[\text { Let } S = 0 . \overline3\]
\[ \Rightarrow S = 0 . 3 + 0 . 03 + 0 . 003 + 0 . 0003 + 0 . 00003 + . . . \infty \]
\[ \Rightarrow S = 0 . 3\left( 1 + {10}^{- 1} + {10}^{- 2} + {10}^{- 3} + {10}^{- 4} + . . . \infty \right)\]
\[\text { S is a geometric series with the first term, a, being 1 and the common ratio, r, being } {10}^{- 1} . \]
\[ \therefore S = \frac{1}{1 - r}\]
\[ \Rightarrow S = 0 . 3\left( \frac{1}{1 - {10}^{- 1}} \right)\]
\[ \Rightarrow S = \frac{3}{9} = \frac{1}{3}\]
RELATED QUESTIONS
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
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:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
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 the following is also in G.P.:
a2, b2, c2
If a, b, c are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
Find the geometric means of the following pairs of number:
2 and 8
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.
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 x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Which term of the G.P. 5, 25, 125, 625, … is 510?
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
The numbers 3, x, and x + 6 form are in G.P. Find nth term
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
0.7, 0.07, 0.007, .....
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Select the correct answer from the given alternative.
If common ratio of the G.P is 5, 5th term is 1875, the first term is -
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:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
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 pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
