Function Challenge!

Algebra Level 3

Define f ( x ) = x 2 − x − 1 2 f(x)=x^2-x-\frac{1}{2} , find the remainder when ∑ x = 1 100 f ( x ) \displaystyle \sum_{x=1}^{100}{f(x)} is divided by 7.

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This is one part of 1+1 is not = to 3 .


The answer is 1.

Kenneth Tan by Kenneth Tan , 21, Malaysia

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1 solution

Kenneth Tan
Aug 11, 2014

∑ x = 1 100 f ( x ) = 1 2 + 2 2 + 3 2 + … + 10 0 2 − 5050 − 50 = 1 2 + 2 2 + 3 2 + … + 10 0 2 − 5100 ≡ 14 ( 1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + 7 2 ) + 9 9 2 + 10 0 2 − 4 ≡ 1 2 + 2 2 − 4 = 1 ( m o d 7 ) \begin{aligned} \displaystyle \sum_{x=1}^{100}{f(x)}&=1^2+2^2+3^2+\ldots+100^2-5050-50\\&=1^2+2^2+3^2+\ldots+100^2-5100\\&\equiv14(1^2+2^2+3^2+4^2+5^2+6^2+7^2)+99^2+100^2-4\\&\equiv1^2+2^2-4\\&=1\pmod{7} \end{aligned} Hence, the remainder when ∑ x = 1 100 f ( x ) \displaystyle \sum_{x=1}^{100}{f(x)} is divided by 7 is 1.

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