Advertisements
Advertisements
प्रश्न
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
Advertisements
उत्तर
a, b, c and d are in G.P.
\[\therefore b^2 = ac\]
\[bc = ad\]
\[ c^2 = bd\] .......(1)
\[\text { LHS } = \frac{ab - cd}{b^2 - c^2}\]
\[ = \frac{ab - cd}{ac - bd} \left[\text { Using } (1) \right]\]
\[ = \frac{\left( ab - cd \right)b}{\left( ac - bd \right)b} \]
\[ = \frac{a b^2 - bcd}{\left( ac - bd \right)b}\]
\[ = \frac{a\left( ac \right) - c\left( c^2 \right)}{\left( ac - bd \right)b} \left[ \text { Using } (1) \right]\]
\[ = \frac{a^2 c - c^3}{\left( ac - bd \right)b}\]
\[ = \frac{c\left( a^2 - c^2 \right)}{\left( ac - bd \right)b}\]
\[ = \frac{\left( a + c \right)\left( ac - c^2 \right)}{\left( ac - bd \right)b}\]
\[ = \frac{\left( a + c \right)\left( ac - bd \right)}{\left( ac - bd \right)b} \left[\text{ Using } (1) \right]\]
\[ = \frac{\left( a + c \right)}{b} = \text { RHS }\]
APPEARS IN
संबंधित प्रश्न
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
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.
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
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 series:
7 + 77 + 777 + ... to n terms;
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
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 \[\frac{1}{r^n}\].
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
If a, b, c, d are in G.P., prove that:
(b + c) (b + d) = (c + a) (c + d)
If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.
If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
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
The value of 91/3 . 91/9 . 91/27 ... upto inf, 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
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
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.
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
Answer the following:
If for a G.P. t3 = `1/3`, t6 = `1/81` find r
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
