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A triangular number is a number that is the sum of the natural numbers up to some $n$. The closed form is $x = \frac{n(n+1)}{2}$. How do I get $n$ on one side? I've been looking at it from every angle, and I can't find out how. Any help?

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Do you know the quadratic formula? –  Tobias Kildetoft Mar 4 at 14:44

2 Answers 2

The answer elsewhere in this thread is of course correct, but for actual calculation there is a simpler answer. We have $x = \frac12(n^2 +n)$ where $x$ is known and we want to find $n$. This is equivalent to $$2x+\frac14 = \left(n+\frac12\right)^2$$ or, neglecting the fractions, which I could justify with a more careful analysis, but won't, $$n\approx \sqrt{2x}.$$

And indeed the formula $$n = \left\lfloor\sqrt{2x}\right\rfloor$$ always gives the correct answer. ($\lfloor\ldots\rfloor$ just means to drop the fraction.)

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You might want to prove that it always gives the correct answer. Nice trick though :) –  Sawarnik Mar 4 at 19:42
    
I thought it might be more fun if I left it as an exercise. –  MJD Mar 5 at 2:23
    
You might want to check the answer by substituting it back in the original formula though, since this is going to give an integer always. Nice trick +1. :) –  Sabyasachi Mar 5 at 9:26

$$n^2+n=2x$$

$$n^2 +n - 2x = 0$$

$$n = \frac{-1 +\sqrt{1+8x}}{2}\text{ provided $n \in \mathbb N,$ otherwise undefined}$$


For anyone unfamiliar, see proof.

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Ah, how did I forget about the quadratic formula? Thanks for the help! –  undo_all Mar 4 at 14:47
    
You're very welcome. :) –  Sabyasachi Mar 4 at 14:48
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This also proves that $1+8x$ must be a perfect square. –  Sawarnik Mar 4 at 19:27
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@Sawarnik yes, that condition is necessary and sufficient, since for all integers $x$, $1+8x$ is odd, and therefore, $\sqrt{1+8x}$ is odd and thus, $n \in \mathbb Z$ –  Sabyasachi Mar 5 at 9:21

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