Definition: A geometric sequence is an ordered list of numbers in which each term is the previous term multiplied by the same nonzero number, the common ratio.
What makes a sequence geometric?
Dividing any term by the one before it gives the same answer. The list 5, 15, 45, 135 has ratio 3. A ratio between 0 and 1 makes the terms shrink, and a negative ratio makes the signs alternate.
What is the formula?
If the first term is a1 and the ratio is r, then an = a1 · rn-1. The exponent is n – 1 because the first term has not been multiplied yet.
Example. A ball bounces to 90 cm, then 72 cm, then 57.6 cm. The ratio is 72 / 90 = 0.8. The fifth term is 90 · 0.84 = 90 · 0.4096 = 36.864 cm.
What is the usual mistake?
People use rn instead of rn-1, or they multiply a1 by r first and then raise the product to a power. The exponent applies to r only. A geometric sequence is an exponential function whose inputs are whole numbers, so its graph curves instead of lining up.
Where to learn it
- Arithmetic and Geometric Sequences as Functions – lesson
- Exponential Growth and Decay Models – lesson