Advertisements
Advertisements
Question
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.
Advertisements
Solution
Let the first term of an A.P. be a and its common difference be d.
\[a_1 + a_2 + a_3 = 15\]
\[ \Rightarrow a + \left( a + d \right) + \left( a + 2d \right) = 15\]
\[ \Rightarrow 3a + 3d = 15 \]
\[ \Rightarrow a + d = 5 . . . . . . . (i)\]
\[\text { Now, according to the question }: \]
\[a + 1, a + d + 3 \text { and }a + 2d + 9 \text { are in G . P } . \]
\[ \Rightarrow \left( a + d + 3 \right)^2 = \left( a + 1 \right)\left( a + 2d + 9 \right)\]
\[ \Rightarrow \left( 5 - d + d + 3 \right)^2 = \left( 5 - d + 1 \right) \left( 5 - d + 2d + 9 \right) \left[ \text { From } (i) \right] \]
\[ \Rightarrow \left( 8 \right)^2 = \left( 6 - d \right)\left( 14 + d \right)\]
\[ \Rightarrow 64 = 84 + 6d - 14d - d^2 \]
\[ \Rightarrow d^2 + 8d - 20 = 0\]
\[ \Rightarrow \left( d - 2 \right)\left( d + 10 \right) = 0\]
\[ \Rightarrow d = 2, - 10\]
\[\text { Now, putting } d = 2, - 10 \text { in equation (i), we get, a } = 3, 15,\text { respectively } . \]
\[\text { Thus, for } a = 3 \text { and }d = 2, \text { the A . P . is } 3, 5, 7 . \]
\[\text { And, for a = 15 and d = - 10, the A . P . is }15 , 5, - 5 . \]
APPEARS IN
RELATED QUESTIONS
For what values of x, the numbers `-2/7, x, -7/2` are in 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.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
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.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 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
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
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.
Express the recurring decimal 0.125125125 ... as a rational number.
Find the rational number whose decimal expansion is `0.4bar23`.
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 are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
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 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 a, b, c are in A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e 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.
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, 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
Mark the correct alternative in the following question:
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 p2R3 : S3 is equal to
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
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
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
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))`
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n 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 ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
