Advertisements
Advertisements
प्रश्न
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
Advertisements
उत्तर
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
संबंधित प्रश्न
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
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.
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 :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
If \[\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}\] (x ≠ 0), then show that a, b, c and d are in G.P.
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
The ratio of the sum of the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
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 are in G.P., prove that the following is also in G.P.:
a3, b3, c3
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 the fifth term of a G.P. is 2, then write the product of its 9 terms.
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.
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.
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
The fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
The numbers 3, x, and x + 6 form are in G.P. Find nth term
For a G.P. If t4 = 16, t9 = 512, find S10
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
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, ...
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_("n" = 1)^oo 0.4^"n"`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
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 ______.
