Advertisements
Advertisements
प्रश्न
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
विकल्प
(a) (2p − q) (p − 2q)
(b) (2p − q) (2q − p)
(c) (2p − q) (p + 2q)
(d) none of these
Advertisements
उत्तर
(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
संबंधित प्रश्न
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
Given a G.P. with a = 729 and 7th term 64, determine S7.
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
If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.
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:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.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 a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
Write the product of n geometric means between two numbers a and b.
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Which term of the G.P. 5, 25, 125, 625, … is 510?
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 n years.
Express the following recurring decimal as a rational number:
`51.0bar(2)`
Find : `sum_("n" = 1)^oo 0.4^"n"`
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
If for a G.P. t3 = `1/3`, t6 = `1/81` find r
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
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 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
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 ______.
