Advertisements
Advertisements
प्रश्न
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
Advertisements
उत्तर
Let the two numbers be a and b.
geometric mean of a and b = `sqrt"ab"`
Given: a + b = `6sqrt"ab"`
`"a"+ "b" + 2sqrt"ab" = 8sqrt"ab"`
`(sqrt"a" + sqrt"b")^2 = 8sqrt"ab"` .......(i)
`"a" + "b" - 2 sqrt"ab" = 4sqrt"ab"`
`(sqrt"a" - sqrt"b")^2 = 4sqrt"ab"` .......(ii)
Dividing equation (i) by (ii), we get
`(sqrt"a" + sqrt"b")^2/(sqrt"a" - sqrt"b")^2 = (8sqrt"ab")/(4sqrt"ab") = 2`
or `(sqrt"a" + sqrt"b")/(sqrt"a" - sqrt"b") = sqrt2/1`
⇒ `((sqrt"a" + sqrt"b") + (sqrt"a" - sqrt"b"))/((sqrt"a" + sqrt"b") - (sqrt"a" - sqrt"b")) = (sqrt2 + 1)/(sqrt2 - 1)`
`(2sqrt"a")/(2sqrt"b") = sqrt"a"/sqrt"b" = (sqrt2 + 1)/(sqrt2 - 1)`
On squaring, `"a"/"b" =(sqrt2 + 1)^2/(sqrt2 - 1)^2 = (3 + 2sqrt2)/(3 - 2sqrt2)`
Hence, `"a"/"b" =(3 + 2sqrt2)/(3 - 2sqrt2)`
APPEARS IN
संबंधित प्रश्न
Which term of the following sequence:
`sqrt3, 3, 3sqrt3`, .... is 729?
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
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`.
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is `1/r^n`.
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 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 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 \[\frac{3069}{512}\] ?
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.
Find the rational number whose decimal expansion is `0.4bar23`.
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.
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:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
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 (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
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.
Write the product of n geometric means between two numbers a and b.
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
For the G.P. if r = `1/3`, a = 9 find t7
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Express the following recurring decimal as a rational number:
`51.0bar(2)`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
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
The sum or difference of two G.P.s, is again a G.P.
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
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 ______.
