Advertisements
Advertisements
प्रश्न
If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
उत्तर
\[\text {It is given that A_1 and A_2 are the A . M . s between a and b } . \]
\[\text{ Thus, a , A_1 , A_2 and b are in A . P . with common difference d }. \]
\[\text{ Here }, d = \frac{b - a}{3}\]
\[ \therefore A_1 = a + \frac{b - a}{3} = \frac{2a + b}{3}\]
\[\text{ and } A_2 = a + \frac{2\left( b - a \right)}{3} = \frac{a + 2b}{3}\]
\[\text{ It is also given that G_1 and G_2 are the G . M . s between a and b } . \]
\[\text{ Thus, a , G_1 , G_2 and b are in G . P . with common ratio r } . \]
\[\text{ Here }, r = \left( \frac{b}{a} \right)^\frac{1}{3} \]
\[ \therefore G_1 = a \left( \frac{b}{a} \right)^\frac{1}{3} = b^\frac{1}{3} a^\frac{1}{3} \]
\[\text{ and } G_2 = a \left[ \left( \frac{b}{a} \right)^\frac{1}{3} \right]^2 = b^\frac{1}{3} a^\frac{1}{3} \]
\[ \Rightarrow \frac{A_1 + A_2}{G_1 G_2} = \frac{\frac{2a + b}{3} + \frac{a + 2b}{3}}{b^\frac{1}{3} a^\frac{1}{3} \times b^\frac{1}{3} a^\frac{1}{3}} = \frac{a + b}{ab}\]
\[\]
APPEARS IN
संबंधित प्रश्न
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
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.
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
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 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
Find the 4th term from the end of the G.P.
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
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.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Find the sum :
\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
If a, b, c are in G.P., prove that log a, log b, log c 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:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
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, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
The two geometric means between the numbers 1 and 64 are
If for a sequence, tn = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
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:
`1/2, 1/4, 1/8, 1/16,...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
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 third term of a G.P. is 4, the product of the first five terms is ______.
The sum or difference of two G.P.s, is again a G.P.
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
The sum of infinite number of terms of a decreasing G.P. is 4 and the sum of the terms to m squares of its terms to infinity is `16/3`, then the G.P. is ______.
