Sums of the first n numbers, squares

Problem: Find the sum of the first 1000 natural numbers. Solution: Write the numbers twice as follows: $ \begin{matrix} 1 & + & 2 & + & \dots & + & 999 & + & 1000 \\\ 1000 & + & 999 & + & \dots & + & 2 & + & 1 \end{matrix}$ Summing the numbers in each column, we have: $ 2 (1 + 2 + \dots + 1000) = 1001 + 1001 + \dots + 1001$ (1000 times)...