Let `lta_(n)gt` be an arithmetic sequence whose first term is `1` and `ltb_(n)gt` be a geometric sequence whose first term is `2`. If the common ratio of geometric sequence is half the common difference of arithmetic sequence, the minimum value of `(a_(4)b_(1) + a_(3)b_(2) + 2a_(1)b_(3))` is equal to

Options

A. `(25)/(12)`
B. `(-3)/(2)`
C. `(3)/(2)`
D. `(-25)/(12)`

Answer

D

Explanation

Terms of `AP` are: `1,1 + d,1 + 2d,1 + 3d`,……
Terms of `GP` are: `2,d,(d^(2))/(2),(d^(3))/(4)`,…….
`:. a_(4)b_(1) + a_(3)b_(2) + 2a_(1)b_(3)`
`= (2 + 6d) + (1 + 2d)d + 2(1 xx (d^(2))/(2))`
`= 3d^(2) + 7d + 2 = 3(d+(7)/(6)) - (25)/(12)`
`:.` Minimum value `= -25//12`