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If terms a(1),a(2),a(3)….,a(50) are in A...

If terms `a_(1),a_(2),a_(3)….,a_(50)` are in A.P and `a_(6)=2`, then the value of common difference at which maximum value of `a_(1)a_(4)a_(5)` occur is

A

`(6)/(5)`

B

`(2)/(3)`

C

`(8)/(5)`

D

`(3)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
`f = a_(1) a_(4) a_(5)`
`a_(6) = 2`
`a + 5d = 2`
`= a (a + 3d) (a + 4d) = (2 - 5d) (2-5d + 3d) (2-5d) + 4d)`
`=(2 - 5d) (2 - 2d) (2-d) = 2(2 - 5d) (2-d - 2d + d^(2))`
`=2(2 - 5d) (d^(2) - 3d + 2) = 2(2d^(2) - 6d + 4- 5d^(3) + 15d^(2) - 10d)`
`=2 (-5d^(3) + 17d^(2) - 16d + 4) = -2 [5d^(3) - 17d^(2) + 16d -4]`
`f' = -2[15d^(2) - 34d + 16]`
`15d^(2) - 34d + 16 = 0`
`15 d^(2) - 24d - 10d + 16 = 0`
`3d (5d -8) (3d - 2 ) = 0`
`d = (8)/(5), (2)/(3)`
`f'' = -2 [30d - 34] = -2 [overset(6)(cancel(30))xx (8)/(cancel5) -34] = -28`
At `d = (8)/(5)` given function is maximum.
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