Question

Let $a_1,a_2,...,a_{100}$ be non-zero real numbers such that $a_1+a_2+...+a_{100}=0$. Then

• A. $\sum _{i=1}^{100}{a}_i{2}^{a_i}>0\text{and}\sum _{i=1}^{100}{a}_i{2}^{-a_i}<0$
• B. $\sum _{i=1}^{100}{a}_i{2}^{a_i}\ge 0\text{and}\sum _{i=1}^{100}{a}_i{2}^{-a_i}\ge 0$
• C. $\sum _{i=1}^{100}{a}_i{2}^{a_i}\le 0\text{and}\sum _{i=1}^{100}{a}_i{2}^{-a_i}\le 0$
• D. the sign of $\sum _{i=1}^{100}{a}_i{2}^{a_i}\text{or}\sum _{i=1}^{100}{a}_i{2}^{-a_i}$ depends on the choice of $a_i\text{'s}$

Answer 1

The correct option is A $\sum _{i=1}^{100}{a}_i{2}^{a_i}>0\text{and}\sum _{i=1}^{100}{a}_i{2}^{-a_i}<0$

$\text{For every real number}{a}_i,a_i{2}^{a_i}>a_i\&a_i{2}^{-a_i}<a_i\Rightarrow \sum _{i=1}^{100}{a}_i{2}^{a_i}>\sum _{i=1}^{100}{a}_i\&\sum _{i=1}^{100}{a}_i{2}^{-a_i}<\sum _{i=1}^{100}{a}_i\Rightarrow \sum _{i=1}^{100}{a}_i{2}^{a_i}>0\&\sum _{i=1}^{100}{a}_i{2}^{-a_i}<0$