Question

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

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

Answer 1

Answer: (A)

We have, $a_1,a_2,a_3,...,a_{100}$ be non-zero real number and

$a_1+a_2+a_3+...+a_{100}=0$

$a_i\cdot 2^{a_i}>a_i$ and $a_i\cdot 2^{-a_i}<a_i$

$\sum _{i=1}^{100}{a}_1\cdot 2^{a_i}>\sum _{i=1}^{100}{a}_i$ and $\sum _{i=1}^{100}{a}_1\cdot 2^{-a_i}<\sum _{i=1}^{100}{a}_i$

$\Rightarrow \sum _{i=1}^{100}{a}_1\cdot 2^{a_i}>0$ and $\sum _{i=1}^{100}{a}_1\cdot 2^{-a_i}<0$