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 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
Evaluate `sum_(k=1)^11 (2+3^k )`
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 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.
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.
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
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.
Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
Find the sum of the following series:
7 + 77 + 777 + ... to n 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.
Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.
If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term is ______.
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
For the G.P. if a = `7/243`, r = 3 find t6.
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 a G.P. if S5 = 1023 , r = 4, Find a
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: `sum_("r" = 1)^10 5 xx 3^"r"`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Find : `sum_("n" = 1)^oo 0.4^"n"`
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 the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
