Advertisements
Advertisements
प्रश्न
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Advertisements
उत्तर
Let the three numbers in G.P. be `"a"/"r", "a", "ar"`.
According to the given conditions,
`"a"/"r" + "a" + "ar"` = 35
∴ `"a"(1/"r" + 1 + "r")` = 35 ....(i)
Also, `("a"/"r")("a")("ar")` = 1000
∴ a3 = 1000
∴ a = 10
Substituting the value of a in (i), we get
`10(1/"r" + 1 + "r")` = 35
∴ `1/"r" + "r" + 1 = 35/10`
∴ `1/"r" + "r" = 35/10 - 1`
∴ `1/"r" + "r" = 25/10`
∴ `1/"r" + "r" = 5/2`
∴ 2r2 – 5r + 2 = 0
∴ (2r – 1) (r – 2) = 0
∴ r = `1/2` or r = 2
When r = `1/2`, a = 10
`"a"/"r" = 10/((1/2))` = 20, a = 10 and ar = `10(1/2)` = 5
When r = 2, a = 10
`"a"/"r" = 10/2` = 5, a = 10 and ar = 10 (2) = 20
Hence, the three numbers in G.P. are 20, 10, 5 or 5, 10, 20.
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.
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
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
Find three numbers in G.P. whose sum is 38 and their product is 1728.
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
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 geometric series:
x3, x5, x7, ... to n terms
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
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.
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
Three numbers are in A.P. and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P. Find the numbers.
If a, b, c are in G.P., prove that the following is also in G.P.:
a3, b3, c3
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
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 \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.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.
Find the geometric means of the following pairs of number:
2 and 8
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
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 10 years.
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the areas of all the squares
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
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:
If for a G.P. t3 = `1/3`, t6 = `1/81` find r
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 ______.
