Advertisements
Advertisements
प्रश्न
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
उत्तर
\[\text { In the given G . P . , first term, } a = 8\]
\[ \text { and common ratio, } r = \frac{1}{\sqrt{2}}\]
\[\text { Hence, the sum S to infinity is given by } S = \frac{a}{1 - r} = \frac{8}{1 - \frac{1}{\sqrt{2}}} = \left( 2 + \sqrt{2} \right) . \]
APPEARS IN
संबंधित प्रश्न
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
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:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If a, b, c are in G.P., then prove that:
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 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.
Find the geometric means of the following pairs of number:
2 and 8
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
If for a sequence, tn = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
For a G.P. a = 2, r = `-2/3`, find S6
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
