Advertisements
Advertisements
प्रश्न
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
उत्तर
geometric progressions 1, –a, a2, –a3,…
First term, a = 1, common ratio, r = `(-"a")/1 = -"a"`
∴ Sum of n terms = `("a"(1 - "r"^"n"))/(1 - "r")`, r > 1
= `("a"(-a)^"n")/(1 - "r")`, r > 1
= `(1.[1 - (-"a")^"n"])/(1 -(-"a"))`
= `([1 - (-a)^"n"])/(1 + "a")`
APPEARS IN
संबंधित प्रश्न
Given a G.P. with a = 729 and 7th term 64, determine S7.
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.
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 `1/r^n`.
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) 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, ...
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Which term of the G.P. :
\[\sqrt{3}, 3, 3\sqrt{3}, . . . \text { is } 729 ?\]
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
If \[\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}\] (x ≠ 0), then show that a, b, c and d are in G.P.
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
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:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
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).
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:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
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
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
For the G.P. if a = `2/3`, t6 = 162, find r.
Which term of the G.P. 5, 25, 125, 625, … is 510?
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third 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.
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...`
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
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:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
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 third term of G.P. is 4. The product of its first 5 terms is ______.
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 ______.
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 ______.
