Geometric Sequences

In a geometric sequence, you get the next term by multiplying the current one by a fixed number called the common ratio, r. Start at g0 and multiply by r again and again: g0, g0 r, g0 r², g0 r³, and so on. The exponent on r always matches how many multiplications you have done, which is one less than the term number, so the nth term is g0 times r^n-1. That is why repeated multiplication looks so different from repeated addition on a graph: an arithmetic sequence climbs by the same flat amount each step, but a geometric one climbs by a percentage of whatever it already is. Fold a sheet of paper in half 20 times and the thickness doubles 20 times over, starting from about 0.1 millimeters: that is 0.1 times 2²⁰, over 100,000 millimeters, taller than most buildings.

Plotted against n, a geometric sequence with r greater than 1 curves upward faster and faster, the signature shape of exponential growth. With 0 < r < 1, the points fall toward the horizontal axis, getting closer and closer to 0 without ever landing on it, since multiplying a positive number by a positive fraction always leaves something positive behind. When r is negative, the terms flip sign every step, so the graph bounces between positive and negative values while still growing or shrinking in size depending on whether absolute value of r is above or below 1.

This is the discrete cousin of the exponential functions coming next in the unit: a geometric sequence is just an exponential function evaluated only at whole numbers n. Seeing the pattern here first makes the continuous exponential formula feel inevitable rather than new.