A problem on Floor and Ceiling Functions!

Is it true or false?

1+1+8n21=2n\large{\left \lceil \dfrac{1 + \sqrt{1+8n}}{2} \right \rceil - 1 = \left \lfloor \sqrt{2n} \right \rfloor}

Here nn is a positive integer.

If it is false, provide me a counter-example. If it's true, please provide me a proof.

Note by Satyajit Mohanty
4 years, 3 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

False. Often, they don't agree.

Michael Mendrin - 4 years, 3 months ago

Log in to reply

I'm sorry. I forgot to mention that nn is a positive integer! Can you find some counter-example now?

Satyajit Mohanty - 4 years, 3 months ago

Log in to reply

..

12(1+1+84)1=2.37228...\dfrac { 1 }{ 2 } (1+\sqrt { 1+8\cdot 4 } )-1=2.37228...
24=2.828427...\sqrt { 2\cdot 4 } =2.828427...

But because of the way ceiling and floor functions work, these two go off in opposite directions. This is just one example.

Michael Mendrin - 4 years, 3 months ago

Log in to reply

if n = 2 it is not true

the only solutions are: 0,1,3,4,6

John Doe - 4 years, 3 months ago

Log in to reply

It is true for n=2n=2. I don't agree!

Satyajit Mohanty - 4 years, 3 months ago

Log in to reply

[] means floor?

John Doe - 4 years, 3 months ago

Log in to reply

@John Doe Look. .\lceil . \rceil means ceiling and .\lfloor . \rfloor means floor.

Satyajit Mohanty - 4 years, 3 months ago

Log in to reply

@Satyajit Mohanty Ah- i see =)

John Doe - 4 years, 3 months ago

Log in to reply

@John Doe It's not true when n = 12 (calculated with Wolfram Alpha)

John Doe - 4 years, 3 months ago

Log in to reply

@Satyajit Mohanty It's not true when n = 12 (calculated with Wolfram Alpha)

John Doe - 4 years, 3 months ago

Log in to reply

Hint: Consider what happens when k22n<(k+1)2 k^2 \leq 2n < (k+1)^2 .

Calvin Lin Staff - 4 years, 3 months ago

Log in to reply

Thanks :) I got it. I actually had a doubt on a solution to this problem: Peculiar Sequence of Positive Integers! as the generalized version of my solution did not match the generalized version of the other solution.

Satyajit Mohanty - 4 years, 3 months ago

Log in to reply

If you see my note, I was pointing out the mistake that @Chew-Seong Cheong made in the generalization.

The true version of the statement that you are looking for, is

1+1+8n2=2n+32{\left \lceil \dfrac{1 + \sqrt{1+8n}}{2} \right \rceil = \left \lfloor \sqrt{2n} + \frac{3}{2} \right \rfloor}

Calvin Lin Staff - 4 years, 3 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...