# Weird!!!

Find the value of $\sum\limits_{x=-2013}^{2013} x \cdot |x| \cdot \lfloor x \rfloor \cdot \text {sgn} (x)$.

Here, $\lfloor x \rfloor$ denotes the greatest integer function or floor function, and $\text {sgn} (x)$ denotes the sign function: 1 if $x$ is positive, 0 if $x$ is zero, and -1 if $x$ is negative.

I have invented this question, and got the answer $\boxed{0}$ (correct me if I'm wrong), but I don't know how to solve this without using graphs. Please help. Thanks. Note by Jaydee Lucero
6 years, 10 months ago

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Note that $\lfloor x \rfloor = x$ as $x$ is an integer, and $|x| \cdot \text{sgn}(x) = x$ for all $x$. So you're basically asking $\displaystyle\sum_{x=-2013}^{2013} x^3$, which is now easy to solve.

- 6 years, 10 months ago

Precise! So 0 is correct answer.

- 6 years, 10 months ago

To evaluate the sum, note that terms of opposite signs cancel, and the only term left is $0^3 = 0.$

- 6 years, 10 months ago

now I see... thanks... :D

- 6 years, 10 months ago

by graph: The above function takes values of x on the X-axis and y=f(x) on the Y axis. this question is basically to plot the graph of f(x)=x^3, here we are only taking the integers values at x-axis and getting correspond y values as integers.So instead of smooth curve we will will be having the points which lies on the curve of f(x)=x^3.

- 6 years, 10 months ago