If `a_(i)gt0` for `i u=1, 2, 3, … ,n` and `a_(1)a_(2)…a_(n)=1,` then the minimum value of `(1+a_(1))(1+a_(2))…(1+a_(n))`, is

Options

A. `2^(n//2)`
B. `2^(n)`
C. `2^(2n)`
D. 1

Answer

B

Explanation

Using `A.MgeG.M`, we have
`(1+a_(1))gesqrt(1xxa_(1))implies1+a_(1)ge2sqrt(a_(1))`
`(1+a_(2))/(2)gesqrt(1xxa_(2))implies1+a_(2)ge2sqrt(a_(2))`
`vdots" "vdots" "vdots`
`(1+a_(n))/(2)lesqrt(1xxa_(n))implies1+a_(n)ge2sqrt(a_(n))`
`:." "(1+a_(1))(1+a_(2))...(1+a_(n))ge2^(n)sqrt(a_(1)a_(2)...a_(n))`
`implies" "(1+a_(1))(1+a_(2))...(1+a_(n))ge2^(n)`