Advertisements
Advertisements
Question
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Solution
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
RELATED QUESTIONS
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
\[\sqrt{7}, \sqrt{21}, 3\sqrt{7}, . . .\text { to n terms }\]
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.
Find the sum of the following series:
0.6 + 0.66 + 0.666 + .... to n terms
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).
How many terms of the G.P. 3, \[\frac{3}{2}, \frac{3}{4}\] ..... are needed to give the sum \[\frac{3069}{512}\] ?
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) 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.
Find the geometric means of the following pairs of number:
−8 and −2
Write the product of n geometric means between two numbers a and b.
The nth term of a G.P. is 128 and the sum of its n terms is 225. If its common ratio is 2, then its first term is
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
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?
For a G.P. a = 2, r = `-2/3`, find S6
If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [(1.08)5 = 1.47]
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
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 p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
The third term of a G.P. is 4, the product of the first five terms is ______.
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
