Advertisements
Advertisements
Question
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to ______.
Options
1 : 1
(Common ratio)n : 1
(First term)2 : (Common ratio)2
None of these
Advertisements
Solution
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to 1 : 1.
Explanation:
Let us take a G.P. with three terms `a/r, a, ar`.
Then S = `a/r + a + ar = (a(r^2 + r + 1))/r`
P = a3
R = `r/a + 1/a + 1/ar`
= `1/a((r^2 + r + 1)/r)`
`(P^2R^3)/"S"^3 = (a^6 * 11/a^3 ((r^2 + r + 1)/r)^3)/(a^3((r^2 + r + 1)/r)^3` = 1
Therefore, the ratio is 1 : 1
APPEARS IN
RELATED QUESTIONS
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
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 and b are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, c, d, form a G.P. Prove that (q + p): (q – p) = 17 : 15.
Find the 4th term from the end of the G.P.
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If a, b, c, d are in G.P., prove that:
(b + c) (b + d) = (c + a) (c + d)
If a, b, c are in G.P., prove that the following is also in G.P.:
a3, b3, c3
If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are in G.P.
If a, b, c are in A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e are in G.P.
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.
If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
The two geometric means between the numbers 1 and 64 are
For the G.P. if a = `7/243`, r = 3 find t6.
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?
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?
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
For a G.P. If t3 = 20 , t6 = 160 , find S7
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
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.
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the perimeters of all the squares
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
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 (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.
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
The sum or difference of two G.P.s, is again a G.P.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
