# Largest Triangular Number less than a Given Natural Number

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?

#### Solutions Collecting From Web of "Largest Triangular Number less than a Given Natural Number"

$\binom{k}{2}\leq n$ is equivalent to $(2k-1)^2 \leq 8n+1$, hence the largest triangular number $\leq n$ is given by $\binom{k}{2}$ with:
$$k = \left\lfloor \frac{1+\sqrt{8n+1}}{2}\right\rfloor.$$