Advertisements
Advertisements
प्रश्न
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
Advertisements
उत्तर
\[\left( a^2 + b^2 + c^2 \right) p^2 - 2\left( ab + bc + cd \right)p + \left( b^2 + c^2 + d^2 \right) \leq 0\]
\[ \Rightarrow \left( a^2 p^2 + b^2 p^2 + c^2 p^2 \right) - 2\left( abp + bcp + cdp \right) + \left( b^2 + c^2 + d^2 \right) \leq 0\]
\[ \Rightarrow \left( a^2 p^2 - 2abp + b^2 \right) + \left( b^2 p^2 - 2bcp + c^2 \right) + \left( c^2 p^2 - 2cdp + d^2 \right) \leq 0\]
\[ \Rightarrow \left( ap - b \right)^2 + \left( bp - c \right)^2 + \left( cp - d \right)^2 \leq 0\]
\[ \Rightarrow \left( ap - b \right)^2 + \left( bp - c \right)^2 + \left( cp - d \right)^2 = 0\]
\[ \Rightarrow \left( ap - b \right)^2 = 0 \]
\[ \Rightarrow p = \frac{b}{a}\]
\[\text { Also }, \left( bp - c \right)^2 = 0 \]
\[ \Rightarrow p = \frac{c}{b}\]
\[\text { Similiarly }, \Rightarrow \left( cp - d \right)^2 = 0 \]
\[ \Rightarrow p = \frac{d}{c}\]
\[ \therefore \frac{b}{a} = \frac{c}{b} = \frac{d}{c}\]
\[\text { Thus, a, b, c and d are in G . P } .\]
संबंधित प्रश्न
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .
Find the value of n so that `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
Find the 4th term from the end of the G.P.
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
Find the sum of the following series:
7 + 77 + 777 + ... to n terms;
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 the following series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
Three numbers are in A.P. and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P. Find the numbers.
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If a, b, c are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If a, b, c are in A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e are in G.P.
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.
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
For the G.P. if a = `2/3`, t6 = 162, find r.
If for a sequence, tn = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
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.
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
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.
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –
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:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
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 ______.
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 ______.
The third term of G.P. is 4. The product of its first 5 terms is ______.
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 ______.
