Intereting Posts

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Characterization of weak convergence in $\ell_\infty$

I want to determine the closest Triangular number a particular natural number is. For example, the first 10 triangular numbers are $1,3,6,10,15,21,28,36,45,55$ and thus, the number $57$ can be written as

$$57=T_{10}+2$$

The number $54$ can be written as

$$54=T_{9}+9\neq T_{10}-1$$

The second part highlights that I am looking for Triangular numbers larger than a particular positive, and not necessarily “closer” in terms of distance from Triangular numbers.

My approach would be this; given the $n$-th Triangular number has the formula

$$T_n=\frac{n(n+1)}{2}=\binom{n+1}{2}$$

If I’m looking for a particular breakdown and close Triangular number, my number, say, $M$ will be of the form

$$M=T_k+r$$

where $0\le r\le k$, and thus

$$2(M-r)=k^2+k$$

And thus

$$k=\frac{-1\pm\sqrt{1+8(M-r)}}{2}$$

I’m lost here in trying to solve, given that $r$ varies. I know that $r$ is less than or equal to $k$ but for sufficiently large $M$, how would I go about finishing solving?

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$\binom{k}{2}\leq n$ is equivalent to $(2k-1)^2 \leq 8n+1$, hence the largest triangular number $\leq n$ is given by $\binom{k}{2}$ with:

$$ k = \left\lfloor \frac{1+\sqrt{8n+1}}{2}\right\rfloor.$$

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