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
\[\left( a^2 + b^2 + c^2 \right) p^2 - 2\left( ab + bc + cd \right)p + \left( b^2 + c^2 + d^2 \right) \leq 0\]
\[ \Rightarrow \left( a^2 p^2 + b^2 p^2 + c^2 p^2 \right) - 2\left( abp + bcp + cdp \right) + \left( b^2 + c^2 + d^2 \right) \leq 0\]
\[ \Rightarrow \left( a^2 p^2 - 2abp + b^2 \right) + \left( b^2 p^2 - 2bcp + c^2 \right) + \left( c^2 p^2 - 2cdp + d^2 \right) \leq 0\]
\[ \Rightarrow \left( ap - b \right)^2 + \left( bp - c \right)^2 + \left( cp - d \right)^2 \leq 0\]
\[ \Rightarrow \left( ap - b \right)^2 + \left( bp - c \right)^2 + \left( cp - d \right)^2 = 0\]
\[ \Rightarrow \left( ap - b \right)^2 = 0 \]
\[ \Rightarrow p = \frac{b}{a}\]
\[\text { Also }, \left( bp - c \right)^2 = 0 \]
\[ \Rightarrow p = \frac{c}{b}\]
\[\text { Similiarly }, \Rightarrow \left( cp - d \right)^2 = 0 \]
\[ \Rightarrow p = \frac{d}{c}\]
\[ \therefore \frac{b}{a} = \frac{c}{b} = \frac{d}{c}\]
\[\text { Thus, a, b, c and d are in G . P } .\]
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.
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`.
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. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
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 \[\frac{1}{r^n}\].
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
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.
Find the geometric means of the following pairs of number:
−8 and −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 A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
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.
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
Express the following recurring decimal as a rational number:
`2.bar(4)`
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to ______.
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 ______.
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 pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
The third term of G.P. is 4. The product of its first 5 terms is ______.
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then 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.
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 ______.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
The sum of infinite number of terms of a decreasing G.P. is 4 and the sum of the terms to m squares of its terms to infinity is `16/3`, then the G.P. is ______.
