Advertisements
Advertisements
Question
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
Advertisements
Solution
Let the first term be a and the common difference be r.
\[\therefore a_1 + a_2 = 5 \]
\[ \Rightarrow a + ar = 5 . . . \left( i \right)\]
\[\text { Also, } a_n = 3\left[ a_{n + 1} + a_{n + 2} + a_{n + 3} + . . . \infty \right] \forall n \in N\]
\[ \Rightarrow a r^{n - 1} = 3 \left[ a r^{n + 1} + a r^{n + 2} + a r^{n + 3} + . . . \infty \right]\]
\[ \Rightarrow a r^{n - 1} = \frac{3a r^n}{1 - r} \]
\[ \Rightarrow 1 - r = 3r\]
\[ \Rightarrow 4r = 1 \]
\[ \Rightarrow r = \frac{1}{4}\]
\[\text { Putting } r = \frac{1}{4} \text { in } \left( i \right): \]
\[a + \frac{a}{4} = 5\]
\[ \Rightarrow 5a = 20 \]
\[ \Rightarrow a = 4\]
\[\text { Thus, the G . P . is } 4, 1, \frac{1}{4}, \frac{1}{16}, . . . \infty . \]
RELATED QUESTIONS
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
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.
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
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:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
Find the sum of the following series:
0.6 + 0.66 + 0.666 + .... to n terms
If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S1 (S2 + S3).
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:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If logxa, ax/2 and logb x are in G.P., then write the value of x.
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 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 for a sequence, tn = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio
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?
The numbers 3, x, and x + 6 form are in G.P. Find nth 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 3 years.
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 n years.
For a G.P. if a = 2, r = 3, Sn = 242 find n
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
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, ...`
Express the following recurring decimal as a rational number:
`51.0bar(2)`
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Find : `sum_("n" = 1)^oo 0.4^"n"`
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
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is ______.
The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
