9.3Geometric Sequences

The H₂O₂ decay data (1.000 M → 0.500 → 0.250 → 0.125 → 0.0625 M over equal 6-hour steps) is a geometric sequence with common ratio ½ sitting inside a chemistry table; recognizing it as one lets a student predict the next entry without recomputing the integrated rate law from scratch.

The cobalt-60 figure spells out 50% remaining after 5.27 y, 25% after 10.54 y, 12.5% after 15.81 y — a geometric sequence with common ratio 1/2 made physically tangible as a shrinking radioactive sample. Algebra §9.4's own worked examples are growth stories (a school's enrollment climbing 4%/year, website hits climbing 2.6%/week); a bouncing-ball decay problem does appear in the chapter, but only as an unworked end-of-chapter review exercise, not a built-out example, and it never gets the name 'common ratio' attached to a physical quantity the way half-life does here.