How fast can you calculate this?

Algebra Level 3

Given f ( x ) = 2016 + 9 x − 2016 − 9 x ∣ x ∣ . f(x) = \dfrac{ \sqrt{2016+9x} - \sqrt{2016-9x} }{|x|}.

Calculate S = f ( 220 ) + f ( − 221 ) + f ( 222 ) + f ( − 223 ) + f ( − 220 ) + f ( 221 ) + f ( − 222 ) + f ( 223 ) + f ( 224 ) S = f(220) + f(-221) + f(222) + f(-223) + f(-220) + f(221) + f(-222) + f(223) + f(224) .


The answer is 0.2834.

Katyusha Le by Katyusha Le , 20, Vietnam

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1 solution

Chris Lewis
May 13, 2019

Note that f ( x ) + f ( − x ) = 0 f(x)+f(-x)=0 , so all terms except the last cancel. Also note that 2016 = 9 × 224 2016=9\times 224 .

S = f ( 224 ) = 2016 + 2016 − 2016 − 2016 224 = 18 × 224 224 = 3 4 7 = 0.283 … S=f(224)=\frac{\sqrt{2016+2016}-\sqrt{2016-2016}}{224}=\frac{\sqrt{18\times 224}}{224}=\frac{3}{4\sqrt7}=\boxed{0.283\ldots}

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