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
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
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 products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the 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.
Find the 4th term from the end of the G.P.
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
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 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:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
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.
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
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.
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., prove that the following is also in G.P.:
a2, b2, c2
If a, b, c are in G.P., then prove that:
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
The two geometric means between the numbers 1 and 64 are
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
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
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
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 nth term
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
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:
Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.
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
The sum or difference of two G.P.s, is again a G.P.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
