**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.

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## Comments

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TopNewestFalse. Often, they don't agree.

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

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..

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

\(\sqrt { 2\cdot 4 } =2.828427...\)

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

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if n = 2 it is not true

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

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

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[] means floor?

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

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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

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

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