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.

#MathProblem #Math

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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`\boxed{123}`	$\boxed{123}$

Comments

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.

Ivan Koswara - 7 years, 7 months ago

Precise! So 0 is correct answer.

Piyushkumar Palan - 7 years, 7 months ago

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

Michael Tang - 7 years, 7 months ago

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

Jaydee Lucero - 7 years, 6 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.

Shubham Namdeo - 7 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$