Копія Geometric Evaluation of Finite Sums
The following three examples demonstrate the formulae for the sum of powers of the first integers. In each of these demonstrations we follow three steps to create convincing proofs. First we construct the sum and visualize it as an area (or volume in the case of the cubes). The second step involves additional constructions such as doubling , tripling or coloring. The third step is completing the rearrangements of the regions and calculating the sums
Example 8. The sum of the first natural numbers
Example 9. The sum of the squares of the first natural numbers More details.
Example 10. The sum of the cubes of the first natural numbers More details.