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
The sum of first three terms of a G.P. is `39/10` and their product is 1. Find the common ratio and the terms.
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
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:
x3, x5, x7, ... to n terms
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
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 \[\frac{1}{r^n}\].
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
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 numbers having the following decimal expansion:
\[0 . 6\overline8\]
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
For the G.P. if r = − 3 and t6 = 1701, find a.
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
The numbers 3, x, and x + 6 form are in G.P. Find nth term
For a G.P. if S5 = 1023 , r = 4, Find a
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Answer the following:
If for a G.P. t3 = `1/3`, t6 = `1/81` find r
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
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 ______.
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to ______.
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
