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)
\[\left( b^2 + c^2 \right)^2 = \left( b^2 \right)^2 + 2 b^2 c^2 + \left( c^2 \right)^2 \]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = \left( ac \right)^2 + b^2 c^2 + b^2 c^2 + \left( bd \right)^2 \left[\text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = a^2 c^2 + a^2 d^2 + b^2 c^2 + b^2 d^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = a^2 \left( c^2 + d^2 \right) + b^2 \left( c^2 + d^2 \right)\]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = \left( a^2 + b^2 \right)\left( c^2 + d^2 \right)\]
\[\text {Therefore, } \left( a^2 + b^2 \right), \left( c^2 + d^2 \right)\text{ and } \left( b^2 + c^2 \right) \text { are also in G . P } .\]
संबंधित प्रश्न
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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
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.
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
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:
x3, x5, x7, ... to n terms
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;
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
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.
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.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
Find the geometric means of the following pairs of number:
a3b and ab3
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 second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
The two geometric means between the numbers 1 and 64 are
For the G.P. if r = `1/3`, a = 9 find t7
For the G.P. if r = − 3 and t6 = 1701, find a.
For the G.P. if a = `2/3`, t6 = 162, find r.
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
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 3 years.
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"`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is ______.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n 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 ______.
