Progression (Sequence and Series)Hard
Question
If a, b, c and p are distinct real numbers such that (a2 + b2 + c2) p2 - 2(ab + bc + cd) p +(b2 + c2 + d2) ≤ 0, then a, b, c, d
Options
A.are in AP
B.are in GP
C.are in HP
D.satisfy ab = cd
Solution
Here, (a2 + b2 + c2 ) p2 - 2(ab + bc + cd) p + (b2 + c2 + d2) ≤ 0
⇒ (a2 p2 - 2abp + b2) + (b2 p2 - abcp + c2) + (c2 p2 - 2cdp + d2) ≤ 0
⇒ (ap - b)2 + (bp - c)2 + (cp - d)2 ≤ 0
(Since, sum of squares is never less than zero)
⇒ Each of the squares is zero
∴ (ap - b)2 = (bp - c)2 = (cp - d)2 = 0
⇒
∴ a, b, c, d are in GP.
⇒ (a2 p2 - 2abp + b2) + (b2 p2 - abcp + c2) + (c2 p2 - 2cdp + d2) ≤ 0
⇒ (ap - b)2 + (bp - c)2 + (cp - d)2 ≤ 0
(Since, sum of squares is never less than zero)
⇒ Each of the squares is zero
∴ (ap - b)2 = (bp - c)2 = (cp - d)2 = 0
⇒
∴ a, b, c, d are in GP.
Create a free account to view solution
View Solution FreeMore Progression (Sequence and Series) Questions
A piece of wire is bent in the shape of a parabola y = kx2 (y-axis vertical) with a bead of mass m on it. The bead can s...The solution of the equation = 43 in the interval (−π, π) are-...If a, b, c, are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the releation...The sum of infinite terms of the progression 1 + 3x + 5x2 + 7x3 + ......(x < 1) is-...If a,b,c are in A.P., thenare in -...