Advertisements
Advertisements
Question
Express the recurring decimal 0.125125125 ... as a rational number.
Advertisements
Solution
\[\text { Let the rational number S be }0 . \overline{125} .\]
\[ \because S = 0 .\overline{ 125} = 0 . 125 + 0 . 000125 + 0 . 000000125 + 0 . 000000000125 + . . . \infty \]
\[ \Rightarrow S = 0 . 125\left[ 1 + {10}^{- 3} + {10}^{- 6} + {10}^{- 9} + . . . \infty \right]\]
\[\text { Clearly, S is a geometric series with the first term, a, being 1 and the common ratio, r, being } {10}^{- 3} . \]
\[ \therefore S = \frac{1}{\left( 1 - r \right)}\]
\[ \Rightarrow S = 0 . 125\left[ \frac{1}{1 - {10}^{- 3}} \right]\]
\[ \Rightarrow S = \frac{125}{999}\]
RELATED QUESTIONS
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
Evaluate `sum_(k=1)^11 (2+3^k )`
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
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`.
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
Find the 4th term from the end of the G.P.
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 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 progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
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 \[\frac{1}{r^n}\].
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 sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.
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:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (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 G.P., then prove that:
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.
If logxa, ax/2 and logb x are in G.P., then write the value of x.
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
The numbers 3, x, and x + 6 form are in G.P. Find x
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
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.
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
