This note works best when paired with two other Darden technical notes: "The IS/LM Model" (UVA-GEM-0126), and a note that directly supports this one, "Determinants of Economic Growth" (UVA-GEM-0202). This note provides a formal mathematical derivation to address recurring questions about the Solow model (see GEM-0202 for details). Using a detailed derivation that starts from the investment equation used in UVA-GEM-0126, it shows why population growth decreases capital per capita. Finally, it argues that productivity growth is the only path for a country to exhibit long-run output growth in per capita terms. Because this note uses mathematical tools to derive the Solow model, it is well beyond what is needed in an MBA program. It simply provides step-by-step derivations of the Solow model for interested readers.