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Select the correct answer from the given alternative. Sum to infinity of a G.P. 5, -52,54,-58,516,... is –

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प्रश्न

Select the correct answer from the given alternative.

Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –

पर्याय

  • 5

  • `-1/2`

  • `10/3`

  • `3/10`

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उत्तर

Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is `10/3`

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  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 2: Sequences and Series - Miscellaneous Exercise 2.1 [पृष्ठ ४१]

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बालभारती Mathematics and Statistics (Arts and Science) Part 2 [English] Standard 11 Maharashtra State Board
पाठ 2 Sequences and Series
Miscellaneous Exercise 2.1 | Q I. (7) | पृष्ठ ४१

संबंधित प्रश्‍न

For what values of x, the numbers  `-2/7, x, -7/2` are in G.P?


Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).


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 10th term of the G.P.

\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]

 


Find the 4th term from the end of the G.P.

\[\frac{2}{27}, \frac{2}{9}, \frac{2}{3}, . . . , 162\]

Which term of the G.P. :

\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]


If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.


Find three numbers in G.P. whose sum is 65 and whose product is 3375.


The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.


Find the sum of the following geometric progression:

2, 6, 18, ... to 7 terms;


Find the sum of the following geometric progression:

(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;


How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?


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).


Let an be the nth term of the G.P. of positive numbers.

Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.


Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.


Find the rational numbers having the following decimal expansion: 

\[0 . \overline3\]


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.:

a3, b3, c3


If a, b, c, d are in G.P., prove that:

(a2 + b2), (b2 + c2), (c2 + d2) are in 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, c, d are in G.P., prove that:

(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.


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 three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.


The fractional value of 2.357 is 


Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals 


Check whether the following sequence is G.P. If so, write tn.

`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...


For the G.P. if r = `1/3`, a = 9 find t7


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


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 3 years.


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 sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.


Answer the following:

If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2   


Answer the following:

If p, q, r, s are in G.P., show that (pn + qn), (qn + rn) , (rn + sn) are also in G.P.


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 ______.


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