That Turn!

Algebra Level 3

Define the function f : R → R f:\mathbb{R}\to\mathbb{R} as f ( x ) = 2 4 x + 2 . f(x)=\dfrac{2}{4^x+2}. Evaluate ∑ n = 1 2000 f ( n 2001 ) \displaystyle\sum_{n=1}^{2000} f\left(\dfrac{n}{2001}\right) .


The answer is 1000.

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Jubayer Nirjhor by Jubayer Nirjhor , 22, Bangladesh

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1 solution

Michael Mendrin
Mar 28, 2015

Once we notice that

2 4 n 2001 + 2 + 2 4 2001 − n 2001 + 2 = 1 \dfrac { 2 }{ { 4 }^{ \frac { n }{ 2001 } }+2 } +\dfrac { 2 }{ { 4 }^{ \frac { 2001-n }{ 2001 } }+2 } =1

the rest follows.

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Moderator note:

Yes. The key thing for problems like these is to simplify f ( x ) + f ( n − x ) f(x) + f(n-x) to some constant.

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