Advertisements
Advertisements
प्रश्न
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Advertisements
उत्तर
Here, a = 0.15 and r \[= \frac{a_2}{a_1} = \frac{0 . 015}{0 . 15} = \frac{1}{10}\] .
\[S_8 = a\left( \frac{1 - r^8}{1 - r} \right) \]
\[ = 0 . 15\left( \frac{1 - \left( \frac{1}{10} \right)^8}{1 - \frac{1}{10}} \right)\]
\[ = 0 . 15\left( \frac{1 - \frac{1}{{10}^8}}{\frac{1}{10}} \right)\]
\[ = \frac{1}{6}\left( 1 - \frac{1}{{10}^8} \right)\]
APPEARS IN
संबंधित प्रश्न
Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
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:
x3, x5, x7, ... to n terms
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
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
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
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 (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
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, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
The two geometric means between the numbers 1 and 64 are
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
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 nth term
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
For a G.P. if S5 = 1023 , r = 4, Find a
Express the following recurring decimal as a rational number:
`2.bar(4)`
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
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
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
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
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 ______.
