Geometric progressions

Fig. 8.6 doubles each stage: 3, 6, 12, 24, 48, 96, …. Stage n has 3 × 2ⁿ⁻¹ squares; recursively, t₁ = 3 and tₙ = 2tₙ₋₁.

Divide each term by the one before and you always get 2: the common ratio r. A sequence with a common ratio is a geometric progression (GP): a, ar, ar², …, with nth term arⁿ⁻¹.

AP or GP?

• an AP adds the same amount each time;
• a GP multiplies by the same amount each time.

The GP soon races ahead: stage 20 of the doubling pattern has over 1.5 million squares.

The ratio can be any number except 0:

• 1, 3, 9, 27, …: r = 3;
• 1, –1, 1, –1, …: r = –1;
• 5, 15/4, 45/16, …: r = 3/4, so it shrinks.

GPs curve. Plot (n, tₙ) for a GP and the points bend away: they do not lie on a straight line.

To test for a GP, divide each term by the one before. If every answer is the same, it is a GP.