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.
\[\therefore a_1 + a_2 = 5 \]
\[ \Rightarrow a + ar = 5 . . . \left( i \right)\]
\[\text { Also, } a_n = 3\left[ a_{n + 1} + a_{n + 2} + a_{n + 3} + . . . \infty \right] \forall n \in N\]
\[ \Rightarrow a r^{n - 1} = 3 \left[ a r^{n + 1} + a r^{n + 2} + a r^{n + 3} + . . . \infty \right]\]
\[ \Rightarrow a r^{n - 1} = \frac{3a r^n}{1 - r} \]
\[ \Rightarrow 1 - r = 3r\]
\[ \Rightarrow 4r = 1 \]
\[ \Rightarrow r = \frac{1}{4}\]
\[\text { Putting } r = \frac{1}{4} \text { in } \left( i \right): \]
\[a + \frac{a}{4} = 5\]
\[ \Rightarrow 5a = 20 \]
\[ \Rightarrow a = 4\]
\[\text { Thus, the G . P . is } 4, 1, \frac{1}{4}, \frac{1}{16}, . . . \infty . \]
APPEARS IN
संबंधित प्रश्न
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
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.
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.
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:
4, −2, 1, −1/2, ...
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
If a, b, c, d are in G.P., prove that:
(a2 + b2), (b2 + c2), (c2 + d2) are in G.P.
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
The fractional value of 2.357 is
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
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term 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
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
The numbers x − 6, 2x and x2 are in G.P. Find x
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
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
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
Answer the following:
If p, q, r, s are in G.P., show that (pn + qn), (qn + rn) , (rn + sn) are also in G.P.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
If `e^((cos^2x + cos^4x + cos^6x + ...∞)log_e2` satisfies the equation t2 – 9t + 8 = 0, then the value of `(2sinx)/(sinx + sqrt(3)cosx)(0 < x ,< π/2)` is ______.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
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 ______.
