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
If x,y,z are in A.P. , then magnitude of its common difference is-...The first term of an A.P. is 2 and common difference is 4. The sum of its 40 terms will be -...The sum of the infinite series 12 + 22 x + 32 x2 + ..... is-...The value of 91/3. 91/9. 91/27... upto ∞, is-...If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the sum of the squares of their reciproca...