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:
`2, 2sqrt2, 4,.... is 128`
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
Find the sum of the products of the corresponding terms of the sequences `2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2`
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`.
The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
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
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.
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
Find three numbers in G.P. whose sum is 38 and their product is 1728.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
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 series:
9 + 99 + 999 + ... to n terms;
Find the sum :
\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
How many terms of the G.P. 3, \[\frac{3}{2}, \frac{3}{4}\] ..... are needed to give the sum \[\frac{3069}{512}\] ?
Express the recurring decimal 0.125125125 ... as a rational number.
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.
Find the geometric means of the following pairs of number:
2 and 8
Find the geometric means of the following pairs of number:
−8 and −2
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
The fractional value of 2.357 is
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?
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
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,...`
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
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 third term of G.P. is 4. The product of its first 5 terms is ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
