Advertisements
Advertisements
प्रश्न
Express the recurring decimal 0.125125125 ... as a rational number.
Advertisements
उत्तर
\[\text { Let the rational number S be }0 . \overline{125} .\]
\[ \because S = 0 .\overline{ 125} = 0 . 125 + 0 . 000125 + 0 . 000000125 + 0 . 000000000125 + . . . \infty \]
\[ \Rightarrow S = 0 . 125\left[ 1 + {10}^{- 3} + {10}^{- 6} + {10}^{- 9} + . . . \infty \right]\]
\[\text { Clearly, S is a geometric series with the first term, a, being 1 and the common ratio, r, being } {10}^{- 3} . \]
\[ \therefore S = \frac{1}{\left( 1 - r \right)}\]
\[ \Rightarrow S = 0 . 125\left[ \frac{1}{1 - {10}^{- 3}} \right]\]
\[ \Rightarrow S = \frac{125}{999}\]
APPEARS IN
संबंधित प्रश्न
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
The sum of first three terms of a G.P. is `39/10` and their product is 1. Find the common ratio and the terms.
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
Find the sum of the products of the corresponding terms of the sequences `2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2`
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.
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 and b are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, c, d, form a G.P. Prove that (q + p): (q – p) = 17 : 15.
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
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.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first 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 \[\frac{1}{r^n}\].
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
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: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
If logxa, ax/2 and logb x are in G.P., then write the value of x.
If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
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
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
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.
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
For a G.P. If t4 = 16, t9 = 512, find S10
A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
Answer the following:
If p, q, r, s are in G.P., show that (pn + qn), (qn + rn) , (rn + sn) are also in G.P.
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
Let A1, A2, A3, .... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = `1/1296` and A2 + A4 = `7/36`, then the value of A6 + A8 + A10 is equal to ______.
