Advertisements
Advertisements
प्रश्न
Express the following recurring decimal as a rational number:
`51.0bar(2)`
Advertisements
उत्तर
`51.0bar(2)` = 51 + 0.02 + 0.002 + 0.0002 + ... ...(1)
These terms after the first term form a G.P. whose first term is a = 0.02 and common ratio = r = 0.1
Since |r| = |0.1| = 0.1 < 1, the sum to infinity of this G.P. exists and
S = `"a"/(1 - "r")`
= `0.02/(1 - 0.1)`
= `0.02/0.9`
= `2/90`
= `1/45`
∴ from (1), `51.0bar(2) = 51 + 1/45`
= `(2295 + 1)/45`
= `2296/45`
APPEARS IN
संबंधित प्रश्न
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, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
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 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;
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n 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 serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
Express the recurring decimal 0.125125125 ... as a rational number.
Find the rational number whose decimal expansion is `0.4bar23`.
If a, b, c, d are in G.P., prove that:
(a2 + b2), (b2 + c2), (c2 + d2) are in G.P.
If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
If logxa, ax/2 and logb x are in G.P., then write the value of x.
Write the product of n geometric means between two numbers a and b.
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
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.
7, 14, 21, 28, …
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
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:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
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 S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to ______.
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 ______.
