Advertisements
Advertisements
Question
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
Advertisements
Solution
a, b, c and d are in G.P.
\[\therefore b^2 = ac\]
\[ad = bc \]
\[ c^2 = bd\] .......(1)
\[\left( b^2 - c^2 \right)^2 = \left( b^2 \right)^2 - 2 b^2 c^2 + \left( c^2 \right)^2 \]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = \left( ac \right)^2 - b^2 c^2 - b^2 c^2 + \left( bd \right)^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = a^2 c^2 - b^2 c^2 - a^2 d^2 + b^2 d^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = c^2 \left( a^2 - b^2 \right) - d^2 \left( a^2 - b^2 \right)\]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = \left( a^2 - b^2 \right)\left( c^2 - d^2 \right)\]
\[\text { Therefore, } \left( a^2 - b^2 \right), \left( b^2 - c^2 \right) \text { and } \left( c^2 - d^2 \right) \text { are also in G . P } .\]
APPEARS IN
RELATED QUESTIONS
Evaluate `sum_(k=1)^11 (2+3^k )`
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
Given a G.P. with a = 729 and 7th term 64, determine S7.
Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio
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 `(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 progression 0.004, 0.02, 0.1, ... is 12.5?
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
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:
x3, x5, x7, ... to n terms
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
Find the sum :
\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]
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 \[\frac{1}{r^n}\].
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
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.
Find the rational number whose decimal expansion is `0.4bar23`.
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c 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 pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
For the G.P. if a = `7/243`, r = 3 find t6.
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn
3, 6, 12, 24, ...
Find : `sum_("n" = 1)^oo 0.4^"n"`
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 -
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
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.
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` 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 ______.
