Advertisements
Advertisements
प्रश्न
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
Advertisements
उत्तर
\[\text { Here, first term, } a = \sqrt{2} \]
\[\text { and common ratio, }r = \frac{1}{2}\]
\[\text { Let the } n^{th} \text { term be } \frac{1}{512\sqrt{2}} . \]
\[ \therefore a_{n =} \frac{1}{512\sqrt{2}}\]
\[ \Rightarrow a r^{n - 1} = \frac{1}{512\sqrt{2}}\]
\[ \Rightarrow \left( \sqrt{2} \right) \left( \frac{1}{2} \right)^{n - 1} = \frac{1}{512\sqrt{2}}\]
\[ \Rightarrow \left( \frac{1}{2} \right)^{n - 1} = \frac{1}{1024}\]
\[ \Rightarrow \left( \frac{1}{2} \right)^{n - 1} = \left( \frac{1}{2} \right)^{10} \]
\[ \Rightarrow n - 1 = 10 \]
\[ \Rightarrow n = 11\]
\[\text { Thus, the } {11}^{th} \text { term of the given G . P . is } \frac{1}{512\sqrt{2}} .\]
संबंधित प्रश्न
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Evaluate `sum_(k=1)^11 (2+3^k )`
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.
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
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.
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
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 0.004, 0.02, 0.1, ... is 12.5?
Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
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 the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is \[\frac{1}{r^n}\].
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
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 the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
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
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
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.
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
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 10 years.
The numbers x − 6, 2x and x2 are in G.P. Find x
For a G.P. If t4 = 16, t9 = 512, find S10
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
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.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is ______.
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 ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
