KVPY2016MathematicsSequences and Series
Let a ₁, a ₂, . a ₁₀₀ be non-zero real numbers such that a ₁+ a ₂+ .+ a ₁₀₀=0 , Then
Options
- A_ j=1 ¹⁰⁰ a_ i 2^ a_ i >0 and _ i=1 ¹⁰⁰ a_ i 2^ -a_ i < 0
- B_ i =1 ¹⁰⁰ a _ i 2^ a _ i 0 and _ i =1 ¹⁰⁰ a _ i 2^ - a _ i 0
- C_ i=1 ¹⁰⁰ a_ i 2^ a_ i 0 and _ i=1 ¹⁰⁰ a_ i 2^ -a_ i 0
- Dthe sign of _ i=1 ¹⁰⁰ a_ i 2^ a_ i or _ j=1 ¹⁰⁰ a_ i 2^ -a_ i depends on the choice of a_ i 's
Correct answer
A. _ j=1 ¹⁰⁰ a_ i 2^ a_ i >0 and _ i=1 ¹⁰⁰ a_ i 2^ -a_ i < 0
Step-by-step solution
Note that for every real number a _ i array ll & a_ i 2^ a_ i >a_ i and a_ i 2^ -a_ i _ i=1 ¹⁰⁰ a_ i and _ i=1 ¹⁰⁰ a_ i 2^ -a_ i > _ i=1 ¹⁰⁰ a_ i array