Advertisements
Advertisements
Question
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
Advertisements
Solution
\[0 . \overline3\]
\[\text { Let } S = 0 . \overline3\]
\[ \Rightarrow S = 0 . 3 + 0 . 03 + 0 . 003 + 0 . 0003 + 0 . 00003 + . . . \infty \]
\[ \Rightarrow S = 0 . 3\left( 1 + {10}^{- 1} + {10}^{- 2} + {10}^{- 3} + {10}^{- 4} + . . . \infty \right)\]
\[\text { S is a geometric series with the first term, a, being 1 and the common ratio, r, being } {10}^{- 1} . \]
\[ \therefore S = \frac{1}{1 - r}\]
\[ \Rightarrow S = 0 . 3\left( \frac{1}{1 - {10}^{- 1}} \right)\]
\[ \Rightarrow S = \frac{3}{9} = \frac{1}{3}\]
APPEARS IN
RELATED QUESTIONS
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Given a G.P. with a = 729 and 7th term 64, determine S7.
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`.
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.
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
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 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
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, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.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:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If a, b, c are in G.P., then prove that:
If a, b, c are in A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e are in G.P.
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 a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
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.
1, –5, 25, –125 …
For the G.P. if r = − 3 and t6 = 1701, find a.
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
The numbers x − 6, 2x and x2 are in G.P. Find nth term
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
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Express the following recurring decimal as a rational number:
`0.bar(7)`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the perimeters of all the squares
Answer the following:
If for a G.P. t3 = `1/3`, t6 = `1/81` find r
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
If `e^((cos^2x + cos^4x + cos^6x + ...∞)log_e2` satisfies the equation t2 – 9t + 8 = 0, then the value of `(2sinx)/(sinx + sqrt(3)cosx)(0 < x ,< π/2)` is ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
