Advertisements
Advertisements
प्रश्न
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
Advertisements
उत्तर
Here, a = p, r = `"q"/"p"`
If `"q"/"p"` < 1, then
∴ Sn = `("a"(1 - "r"^"n"))/(1 - "r")`
= `("p"[1 - ("q"/"p")^"n"])/(1 - ("q"/"p")`
= `"p"^2/("p" - "q") [1 - ("q"/"p")^"n"]`
If `"q"/"p" > 1,` then
Sn = `("a"("r"^"n" - 1))/("r" - 1)`
= `("p"[("q"/"p")^"n" - 1])/(("q"/"p") - 1)`
= `"p"^2/("q" - "p") [("q"/"p")^"n" - 1]`
APPEARS IN
संबंधित प्रश्न
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
The sum of first three terms of a G.P. is `39/10` and their product is 1. Find the common ratio and the terms.
Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
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.
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
Find the sum of the following series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
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 in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
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 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 a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
The fractional value of 2.357 is
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
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. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
For a G.P. If t4 = 16, t9 = 512, find S10
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
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.
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
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 ______.
The sum or difference of two G.P.s, is again a G.P.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
