Advertisements
Advertisements
Question
Find the rational number whose decimal expansion is `0.4bar23`.
Advertisements
Solution
`S_∞ = a/(1-r)`
S = `0.4bar23 => S = 0.4 + 0.023 + 0.00023 + ... ∞`
`S = 0.4 + 23/10^3 + 23/10^5 + ... ∞`
`S = 0.4 + 23/10^3 {1 + 10^-2 + 10^-4 + ... ∞}`
`S = 0.4 + 23/10^3 {1/(1-1/100)}`
`S = 0.4 + 23/10^3 {100/100-1}`
`S = 0.4 + 23/10^3 xx 100/99`
`S = 4/10 + 23/990`
`S = 419/990`
RELATED QUESTIONS
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
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`
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
If a, b, c are in A.P,; b, c, d are in G.P and ` 1/c, 1/d,1/e` are in A.P. prove that a, c, e are in G.P.
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
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.
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
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.
If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
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:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are 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 \[a^{b - c} b^{c - a} c^{a - b} = 1\]
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
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 the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. 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
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For a G.P. if S5 = 1023 , r = 4, Find a
For a G.P. if a = 2, r = 3, Sn = 242 find n
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
Express the following recurring decimal as a rational number:
`2.3bar(5)`
Find : `sum_("n" = 1)^oo 0.4^"n"`
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
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
