Advertisements
Advertisements
Question
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Advertisements
Solution
Let the required G.P. be a, ar, ar2, ar3, …
Sum to infinity of this G.P. = 5
∴ 5 = `"a"/(1 - "r")`
∴ a = 5(1 – r) ...(i)
Also, the sum of the squares of the terms is 15.
∴ (a2 + a2r2 + a2r4 + …) = 15
∴ 15 = `"a"^2/(1 - "r"^2)`
∴ 15 (1 – r2) = a2
∴ 15(1 – r)(1 + r) = 25 (1 – r)2 ...[From (i)]
∴ 3 (1 + r) = 5 (1 – r)
∴ 3 + 3r = 5 – 5r
∴ 8r = 2
∴ r = `1/4`
∴ a = `5(1 - 1/4) = 5(3/4) = 15/4`
∴ Required G.P. is a, ar, ar2, ar3, …
i.e., `15/4, 15/16, 15/64, ...`
APPEARS IN
RELATED QUESTIONS
For what values of x, the numbers `-2/7, x, -7/2` 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.
Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
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.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) 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:
(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
Find the sum of the following geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
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 (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio 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
The fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5th hour?
The numbers 3, x, and x + 6 form are in G.P. Find x
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
The numbers x − 6, 2x and x2 are in G.P. Find x
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Express the following recurring decimal as a rational number:
`2.3bar(5)`
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
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 –
Select the correct answer from the given alternative.
Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively. (a, b > 0)
Answer the following:
If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
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.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
