If `a b^2c^3, a^2b^3c^4,a^3b^4c^5` are in A.P. `(a ,b ,c >0),` then the minimum value of `a+b+c` is `1` `3` `5` `9`

Options

A. 1
B. 3
C. 5
D. 9

Answer

B

Explanation

A.M. `ge` G.M
`implies (a + b+ c)/(3) ge (abc)^(1//3)`
or `a + b + c ge 3 (abc)^(1//3)`
But given `ab^(2) c^(3), a^(2) b^(3) c^(4), a^(3) b^(4) c^(5)` are in A.P (`:' abc =! 0`) hence,
`2abc = 1 a^(2) b^(2) c^(2)`
`implies (abc - 1)^(2) = 0`
`:. abc = 1`
Now from Eq. (1) we, get
`a + b + c ge 3 (1)^(1//3)`
or `(a + b + c) ge 3`
Hence, minimum value of `a + b + c` is 3.