What the common ratio does.

Bars of the G.P. a, ar, ar², … with the ratio r adjustable through negative and fractional values, showing growth, decay and alternating signs; two-term fit mode finds a and r.

Make the terms alternate in sign and shrink in size (−1 < r < 0); then in fit mode set a₃ = 12, a₆ = 96 and read r = 2, a = 3.

Taking r as first term ÷ second; writing the nth term as arⁿ.

• A geometric progression (G.P.) is a sequence of non-zero terms in which each term divided by the one before it gives the same number r, the common ratio: aₖ₊₁/aₖ = r for every k ≥ 1.
• With first term a, a G.P. reads a, ar, ar², ar³, …. Examples: 3, 6, 12, 24, … (r = 2); 81, −27, 9, −3, … (r = −1/3); 0.2, 0.02, 0.002, … (r = 0.1).
• Compare an arithmetic progression (A.P.), met in earlier classes: there a fixed number d is added each time, as in 100, 150, 200, … with d = 50. A G.P. multiplies by r instead of adding d.
• A negative r makes the signs alternate. A ratio between −1 and 1 makes the terms shrink in size; a ratio above 1 makes them grow fast.
• Notation for G.P. formulas: a = first term, r = common ratio, l = last term, n = number of terms, Sₙ = sum of the first n terms.

Teaching this idea? Put the simulation on your own page or LMS with this code.