Advertisements
Advertisements
Question
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
Options
(a) (2p − q) (p − 2q)
(b) (2p − q) (2q − p)
(c) (2p − q) (p + 2q)
(d) none of these
Advertisements
Solution
(a) (2p − q) (p − 2q)
\[\text{ Let the two numbers be a and b } . \]
\[\text{ a, p, q and b are in A . P } . \]
\[ \therefore p - a = q - p = b - q \]
\[ \Rightarrow p - a = q - p \text{ and } q - p = b - q\]
\[ \Rightarrow a = 2p - q \text{ and } b = 2q - p (i)\]
\[\text{ Also, a, G and b are in G . P }. \]
\[ \therefore G^2 = ab\]
\[ \Rightarrow G^2 = \left( 2p - q \right)\left( 2q - p \right)\]
APPEARS IN
RELATED QUESTIONS
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 pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
Find the value of n so that `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
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.
Find the sum of the following series:
7 + 77 + 777 + ... to n terms;
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.
Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.
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`
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
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 a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (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 the fifth term of a G.P. is 2, then write the product of its 9 terms.
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 the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of 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.
2, 6, 18, 54, …
For the G.P. if r = − 3 and t6 = 1701, find a.
For the G.P. if a = `2/3`, t6 = 162, find r.
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
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.
For a G.P. if a = 2, r = 3, Sn = 242 find n
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
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, ...`
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
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:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
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
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 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 ______.
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
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
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 ______.
