Advertisements
Advertisements
प्रश्न
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Advertisements
उत्तर
Here, a = \[\sqrt{2}\] and r = \[\frac{1}{2}\] .
\[S_8 = a\left( \frac{1 - r^8}{1 - r} \right)\]
\[ = \sqrt{2}\left( \frac{1 - \left( \frac{1}{2} \right)^8}{1 - \frac{1}{2}} \right)\]
\[ = \sqrt{2}\left( \frac{1 - \frac{1}{256}}{\frac{1}{2}} \right)\]
\[ = 2\sqrt{2}\left( \frac{255}{256} \right)\]
\[ = \frac{255\sqrt{2}}{128}\]
APPEARS IN
संबंधित प्रश्न
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Given a G.P. with a = 729 and 7th term 64, determine S7.
If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Find the 4th term from the end of the G.P.
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
Find the sum of the following geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S1 (S2 + S3).
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
If a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
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 a, b, c are in A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e 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 \[a^{b - c} b^{c - a} c^{a - b} = 1\]
The fractional value of 2.357 is
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
For a G.P. If t4 = 16, t9 = 512, find S10
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
Express the following recurring decimal as a rational number:
`0.bar(7)`
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.
If common ratio of the G.P is 5, 5th term is 1875, the first term is -
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
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 for a G.P. first term is (27)2 and seventh term is (8)2, find S8
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:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
