Advertisements
Advertisements
Question
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
Advertisements
Solution
Let the three numbers in G. P. be `"a"/"r"`, a, ar.
According to the given conditions,
`"a"/"r" + "a" + "ar"` = 21
∴ `1/"r" + 1 + "r" = 21/"a"`
∴ `1/"r" + "r" = 21/"a" - 1` ...(i)
Also, `"a"^2/"r"^2 + "a"^2 + "a"^2"r"^2` = 189
∴ `1/"r"^2 + 1 + "r"^2 = 189/"a"^2`
∴ `1/"r"^2 + "r"^2 = 189/"a"^2 - 1` ...(ii)
On squaring equation (i), we get
∴ `1/"r"^2 + "r"^2 + 2 = 441/"a"^2 - 42/"a" + 1`
∴ `(189/"a"^2 - 1) + 2 = 441/"a"^2 - 42/"a" + 1` ...[From (ii)]
∴ `189/"a"^2 + 1 = 441/"a"^2 - 42/"a" + 1`
∴ `441/"a"^2 - 189/"a"^2 - 42/"a"` = 0
∴ `252/"a"^2 = 42/"a"`
∴ 252 = 42a
∴ a = 6
Substituting the value of a in (i), we get
`1/"r" + "r" = 21/6 - 1`
∴ `(1 + "r"^2)/"r" = 15/6`
∴ `(1 + "r"^2)/"r" = 5/2`
∴ 2r2 – 5r + 2 = 0
∴ 2r2 – 4r – r + 2 = 0
∴ (2r – 1) (r – 2) = 0
∴ r = `1/2` or 2.
When a = 6, r = `1/2`,
`"a"/"r"` = 12, a = 6, ar = 3
When a = 6, r = 2
`"a"/"r"` = 3, a = 6, ar = 12
∴ the three numbers are 12, 6, 3 or 3, 6, 12.
APPEARS IN
RELATED QUESTIONS
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
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.
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 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.
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
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?
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
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 \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 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 serie:
5 + 55 + 555 + ... to n terms;
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
Find the geometric means of the following pairs of number:
2 and 8
If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
The fractional value of 2.357 is
For the G.P. if r = − 3 and t6 = 1701, find a.
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
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
3, 6, 12, 24, ...
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
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" = 0)^oo (-8)(-1/2)^"r"`
A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
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
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
Let A1, A2, A3, .... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = `1/1296` and A2 + A4 = `7/36`, then the value of A6 + A8 + A10 is equal to ______.
