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

$\displaystyle\sum_{i = 1}^{100} a_{1}\cdot 2^{a_{i}} > \displaystyle\sum_{i=1}^{100} a_{i} $ and $ \displaystyle\sum _{i=1}^{100} a_{1}\cdot2^{-a_{i}} < \displaystyle\sum _{i=1}^{100} a_{i} $

$ \Rightarrow \displaystyle\sum _{i=1}^{100} a_{1} \cdot 2^{a_{i}} > 0$ and $\displaystyle\sum _{i=1}^{100} a_{1} \cdot 2^{-a_{i}} <0$