Consider the right angled equilateral triangle with the right angle at $(0,0)$. The $i$th row for $0leq i leq n-1$ has $n-i$ points (suppose we denote by balls).

The area of the triangle is supposed to be $sum_{1}^{n} {i}$ but we get the area as $frac{n^2}{2}$ which is different. I don’t understand the fallacy here.