A problem on Floor and Ceiling Functions!

Is it true or false?

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

Here $n$ is a positive integer.

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

#Algebra #GreatestIntegerFunction(Floor) #CeilingFunction #FloorFunction

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Comments

False. Often, they don't agree.

Michael Mendrin - 5 years, 10 months ago

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

Satyajit Mohanty - 5 years, 10 months ago

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

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

Michael Mendrin - 5 years, 10 months ago

if n = 2 it is not true

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