Advertisements
Advertisements
प्रश्न
Find : `sum_("n" = 1)^oo 0.4^"n"`
Advertisements
उत्तर
`sum_("n" = 1)^oo 0.4^"n"`
= 0.4 + (0.4)2 + (0.4)3 + …
The terms 0.4, (0.4)2, (0.4)3 are in G.P.
∴ a = 0.4, r = 0.4
Since, |r| = |0.4| < 1
∴ sum to infinity exists.
∴ `sum_("n" = 1)^oo 0.4^"n" = 0.4/(1 - 0.4)`
= `0.4/0.6`
= `2/3`
APPEARS IN
संबंधित प्रश्न
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`.
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
Find the 4th term from the end of the G.P.
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:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
How many terms of the G.P. 3, \[\frac{3}{2}, \frac{3}{4}\] ..... are needed to give the sum \[\frac{3069}{512}\] ?
Express the recurring decimal 0.125125125 ... as a rational number.
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
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.
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
Find the geometric means of the following pairs of number:
−8 and −2
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after n years.
For a G.P. if S5 = 1023 , r = 4, Find a
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...`
Express the following recurring decimal as a rational number:
`2.3bar(5)`
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
The third term of G.P. is 4. The product of its first 5 terms is ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
