Advertisements
Advertisements
प्रश्न
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
Advertisements
उत्तर
a, b, c and d are in G.P.
\[\therefore b^2 = ac\]
\[ad = bc \]
\[ c^2 = bd\] .......(1)
\[\left( \frac{1}{b^2 + c^2} \right)^2 = \left( \frac{1}{b^2} \right)^2 + \frac{2}{b^2 c^2} + \left( \frac{1}{c^2} \right)^2 \]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \left( \frac{1}{ac} \right)^2 + \frac{1}{b^2 c^2} + \frac{1}{b^2 c^2} + \left( \frac{1}{bd} \right)^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \frac{1}{a^2 c^2} + \frac{1}{a^2 d^2} + \frac{1}{b^2 c^2} + \frac{1}{b^2 d^2} \left[ \text { Using }(1) \right]\]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \frac{1}{a^2}\left( \frac{1}{c^2} + \frac{1}{d^2} \right) + \frac{1}{b^2}\left( \frac{1}{c^2} + \frac{1}{d^2} \right)\]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \left( \frac{1}{a^2 + b^2} \right)\left( \frac{1}{c^2} + \frac{1}{d^2} \right)\]
\[\text{ Therefore }, \left( \frac{1}{b^2 + c^2} \right), \left( \frac{1}{c^2 + d^2} \right)\text { and } \left( \frac{1}{b^2 + c^2} \right) \text { are also in G . P } .\]
APPEARS IN
संबंधित प्रश्न
Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.
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.
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.
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
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 series:
7 + 77 + 777 + ... to n terms;
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of 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.
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.
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
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:
(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.
Find the geometric means of the following pairs of number:
−8 and −2
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If logxa, ax/2 and logb x are in G.P., then write the value of x.
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
For the G.P. if a = `7/243`, r = 3 find t6.
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
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 x − 6, 2x and x2 are in G.P. Find 1st term
For a G.P. If t3 = 20 , t6 = 160 , find S7
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
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.
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Select the correct answer from the given alternative.
If common ratio of the G.P is 5, 5th term is 1875, the first term is -
Answer the following:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
The sum or difference of two G.P.s, is again a G.P.
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 ______.
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 ______.
