Advertisements
Advertisements
प्रश्न
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
Advertisements
उत्तर
\[0 . \overline {231 }\]
\[\text { Let } S = 0 . \overline {231 }\]
\[ \Rightarrow S = 0 . 231 + 0 . 000231 + 0 . 000000231 + . . . \infty \]
\[ \Rightarrow S = 0 . 231\left( 1 + {10}^{- 3} + {10}^{- 6} + . . . \infty \right)\]
\[\text { It is a G . P } . \]
\[ \therefore S = 0 . 231\left( \frac{1}{1 - {10}^{- 3}} \right)\]
\[ \Rightarrow S = \frac{231}{999}\]
APPEARS IN
संबंधित प्रश्न
Which term of the following sequence:
`sqrt3, 3, 3sqrt3`, .... is 729?
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
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`?
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
Find the sum of the following series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
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.
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.
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 the following is also in G.P.:
a3, b3, c3
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.
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Find the geometric means of the following pairs of number:
a3b and ab3
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
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.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
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
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
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:
For a sequence Sn = 4(7n – 1) verify that the sequence is 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.
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 ______.
