Advertisements
Advertisements
प्रश्न
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
उत्तर
Let the first term be a and the common difference be r.
\[\therefore a_1 + a_2 = 5 \]
\[ \Rightarrow a + ar = 5 . . . \left( i \right)\]
\[\text { Also, } a_n = 3\left[ a_{n + 1} + a_{n + 2} + a_{n + 3} + . . . \infty \right] \forall n \in N\]
\[ \Rightarrow a r^{n - 1} = 3 \left[ a r^{n + 1} + a r^{n + 2} + a r^{n + 3} + . . . \infty \right]\]
\[ \Rightarrow a r^{n - 1} = \frac{3a r^n}{1 - r} \]
\[ \Rightarrow 1 - r = 3r\]
\[ \Rightarrow 4r = 1 \]
\[ \Rightarrow r = \frac{1}{4}\]
\[\text { Putting } r = \frac{1}{4} \text { in } \left( i \right): \]
\[a + \frac{a}{4} = 5\]
\[ \Rightarrow 5a = 20 \]
\[ \Rightarrow a = 4\]
\[\text { Thus, the G . P . is } 4, 1, \frac{1}{4}, \frac{1}{16}, . . . \infty . \]
संबंधित प्रश्न
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
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
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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
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 series:
7 + 77 + 777 + ... to n terms;
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is \[\frac{1}{r^n}\].
Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.
The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.
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, d are in G.P., prove that:
(b + c) (b + d) = (c + a) (c + d)
If a, b, c are in G.P., then prove that:
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a 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 pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
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
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
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
For a G.P. if S5 = 1023 , r = 4, Find a
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
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
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:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
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 ______.
