Advertisements
Advertisements
प्रश्न
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) 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)
\[\left( b^2 - c^2 \right)^2 = \left( b^2 \right)^2 - 2 b^2 c^2 + \left( c^2 \right)^2 \]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = \left( ac \right)^2 - b^2 c^2 - b^2 c^2 + \left( bd \right)^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = a^2 c^2 - b^2 c^2 - a^2 d^2 + b^2 d^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = c^2 \left( a^2 - b^2 \right) - d^2 \left( a^2 - b^2 \right)\]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = \left( a^2 - b^2 \right)\left( c^2 - d^2 \right)\]
\[\text { Therefore, } \left( a^2 - b^2 \right), \left( b^2 - c^2 \right) \text { and } \left( c^2 - d^2 \right) \text { are also in G . P } .\]
संबंधित प्रश्न
Which term of the following sequence:
`sqrt3, 3, 3sqrt3`, .... is 729?
Find the sum of the products of the corresponding terms of the sequences `2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2`
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
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 a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .
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.
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 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Find three numbers in G.P. whose sum is 38 and their product is 1728.
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 geometric progression:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
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.
If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
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:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
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.
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 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
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term is ______.
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
The two geometric means between the numbers 1 and 64 are
For the G.P. if r = `1/3`, a = 9 find t7
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 the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
For a G.P. if S5 = 1023 , r = 4, Find a
For a G.P. If t3 = 20 , t6 = 160 , find S7
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is 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 ______.
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 ______.
