Advertisements
Advertisements
प्रश्न
If a, b, c, d are in G.P., prove that:
(b + c) (b + d) = (c + a) (c + d)
उत्तर
a, b, c and d are in G.P.
\[\therefore b^2 = ac\]
\[bc = ad\]
\[ c^2 = bd\] .......(1)
\[\text { LHS } = \left( b + c \right)\left( b + d \right)\]
\[ = b^2 + bd + bc + cd\]
\[ = ac + c^2 + ad + cd \left[ \text { Using } (1) \right]\]
\[ = c\left( a + c \right) + d\left( a + c \right)\]
\[ = \left( c + a \right)\left( c + d \right) =\text { RHS }\]
APPEARS IN
संबंधित प्रश्न
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.
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`.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
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.
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Find the sum of the following series:
7 + 77 + 777 + ... to n terms;
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
Find the sum of the following serie to infinity:
\[\frac{1}{3} + \frac{1}{5^2} + \frac{1}{3^3} + \frac{1}{5^4} + \frac{1}{3^5} + \frac{1}{56} + . . . \infty\]
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
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.
Write the product of n geometric means between two numbers a and b.
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
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 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. a = 2, r = `-2/3`, find S6
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
For a G.P. If t3 = 20 , t6 = 160 , find S7
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.
Select the correct answer from the given alternative.
Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively. (a, b > 0)
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is ______.
The third term of a G.P. is 4, the product of the first five terms 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 ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
