Which Number Falls In the Category?

$\begin{aligned} & 1 \\& 2 \quad 3 \\& 4 \quad 5 \quad 6 \\& 7\quad 8 \quad 9 \quad 10 \\& \vdots\end{aligned}$ We note that the numbers in the each row is equal to rowth number and every row ends up with sum of $r^{th}$ rows.

Let the $r_j$ be the any row and $2010$ should also fall in any of the $r_j$ row number. Then we note that $\begin{aligned}&\displaystyle\sum_{r= 1}^{j-1} r < 2010 \leq \displaystyle\sum_{r=1 }^{j}r\\& 1953 < 2010 \leq 2016 \\& \displaystyle\sum_{r=1 }^{63-1} r < 2010 \leq \displaystyle\sum_{r=1}^{63} r \end{aligned}$ Shows that $2010$ falls in $63^{rd}$ row which has 63 distinct number from 1954 to 2016 and 2010 is at 7th position before from back ( 2016 ) or 56th position after from 1954.

Now the number exactly below the 2010 is in the 64th row at the 56th position after from 2017 and 7th position back from 2080 .

Therefore the number just below 2010 will be $2080 -7 =\boxed{2073}$ .