Let `a_1, a_2, a_100` be non-zero real numbers such that `a_1 +a_2 + +a_(100) = 0.` Then

Options

A. `sumunderset(i=l)overset(100)(a_(i)2^(a_(i))) gt and sumunderset(i=l)overset(100)(a_(i)2^(-a_(i)))lt0`
B. `sumunderset(i=l)overset(100)(a_(i)2^(a_(i))) ge and sumunderset(i=l)overset(100)(a_(i)2^(-a_(i)))ge0`
C. `sumunderset(i=l)overset(100)(a_(i)2^(a_(i))) le and sumunderset(i=l)overset(100)(a_(i)2^(-a_(i)))le0`
D. the sign of `sumunderset(i=l)overset(100)(a_(i)2^(a_(i))) orsumunderset(i=l)overset(100)(a_(i)2^(-a_(i)))` depends on the choice of `a_(i)` 's

Answer

a

Explanation

Note that for every real number `a_(i)`
`a_(i)2^(ai)gta_(i) and a_(i) 2^(-a_(i))lta_(i)`
therefore, `overset(100)underset(i-l)suma_(i).2^(a_(i))gtoverset(100)underset(i-l)suma_(i) and overset(100)underset(i-l)suma_(i).,2^(-a)gtoverset(100)underset(i-l)suma_(i)`