Advertisements
Advertisements
Question
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.
Advertisements
Solution
Let the terms of the given G.P. be \[\frac{a}{r}, \text { a and ar }\]
∴ Product = 216
\[\Rightarrow a^3 = 216\]
\[ \Rightarrow a = 6\]
It is given that \[\frac{a}{r} + 2, a + 8 \text { and ar } + 6\] are in A.P.
\[\therefore 2\left( a + 8 \right) = \frac{a}{r} + 2 + ar + 6\]
\[\text { Putting a = 6, we get }\]
\[ \Rightarrow 28 = \frac{6}{r} + 2 + 6r + 6\]
\[ \Rightarrow 28r = 6 r^2 + 8r + 6\]
\[ \Rightarrow 6 r^2 - 20r + 6 = 0\]
\[ \Rightarrow \left( 6r - 2 \right)\left( r - 3 \right) = 0\]
\[ \Rightarrow r = \frac{1}{3}, 3\]
\[\text { Hence, putting the values of a and r, the required numbers are 18, 6, 2 or 2, 6 and 18 }.\]
APPEARS IN
RELATED QUESTIONS
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.
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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
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 the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
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.
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.
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.
If S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If a, b, c are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
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., 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 (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
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 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
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
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 3 years.
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
For a G.P. If t3 = 20 , t6 = 160 , find S7
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...`
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
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
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 perimeters of all the squares
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
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.
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 ______.
