Advertisements
Advertisements
Question
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
Advertisements
Solution
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to `a/b` or `b/c`.
Explanation:
Since a, b and c are in G.P
∴ `b/a = c/b` = r .....(Constant)
⇒ b = ar and c = br
⇒ c = ar · r = ar2
So `(a - b)/(b - c) = (a - ar)/(ar - ar^2)`
= `(a(1 - r))/(ar(1 - r))`
= `1/r`
= `a/b`
= `b/c`
APPEARS IN
RELATED QUESTIONS
Given a G.P. with a = 729 and 7th term 64, determine S7.
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
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.
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 `1/r^n`.
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.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
Find the sum :
\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
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 numbers having the following decimal expansion:
\[0 . 6\overline8\]
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 A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
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 S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
For the G.P. if a = `7/243`, r = 3 find t6.
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
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
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
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:
`-3, 1, (-1)/3, 1/9, ...`
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, ...`
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Find : `sum_("n" = 1)^oo 0.4^"n"`
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
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 -
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.
The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
