Waste less time on Facebook — follow Brilliant.
×

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
4 years ago

No vote yet
3 votes

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} \( \boxed{123} \)

Comments

Sort by:

Top Newest

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 - 4 years ago

Log in to reply

Precise! So 0 is correct answer.

Piyushkumar Palan - 4 years ago

Log in to reply

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

Michael Tang - 4 years ago

Log in to reply

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

Jaydee Lucero - 4 years ago

Log in to reply

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 - 4 years ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...