Abstract
By counting blocks within blocks, we geometrically derive several basic additive identities. More precisely, we consider a cube with edge length n,for any positive integer n.This cube may be envisioned as being built by n 3 cubes of unit edge length. Focusing upon these latter ‘building blocks’, we count the number of cubes, square based parallelepipeds and arbitrary parallelepipeds (boxes) that are formed by these blocks within the larger cube. By taking two approaches in counting the number of these geometric solids, formulas are found for the sum of the cubes, the fourth powers and the fifth powers, respectively, of the first npositive integers.
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