Geometric Sequences — and the Doorway to Chapter 14

In this lesson you will: Find the common ratio and use it to write recursive and explicit rules; Translate a geometric sequence between recursive and explicit form; Decide whether a list or a situation is arithmetic, geometric, or neither, and build a model from it.

Multiply instead of add

A sequence is geometric when each term is the term before it multiplied by the same number. That number is the common ratio, written r, and you find it by dividing: r=(a_n)/(a_(n-1)).

For 3,6,12,24,48 the ratios are 6/3=2, (12)/(6)=2, (24)/(12)=2 and (48)/(24)=2. The differences, by contrast, are 3,6,12,24 — not constant, which is exactly why this is not arithmetic.

The exponent counts steps, not terms

Each step multiplies by r, and getting from a_1 to a_n takes n-1 steps, so a_n carries n-1 factors of r.

This is the arithmetic off-by-one again, relocated into an exponent where it is harder to see. The same cure works: test the rule at n=1, where r^0=1 should hand back a_1 untouched.

Which kind is it?

Chapter 9 sorted linear from exponential models by asking whether a quantity changes by a constant amount or a constant factor. Sequences are the same question in list form: look at consecutive terms and ask what stays the same.

Do both tests before deciding. Differences first, then ratios. Plenty of sequences pass neither, and \"neither\" is a legitimate answer.

Common mistake — A recursive rule with nothing to start it

Asked to describe 2,6,18,54,... recursively, a student writes a_n=3a_(n-1) and puts down the pencil.

The rule is not wrong. It is incomplete, and incompleteness is harder to notice than an error, because the line you did write is correct and looks like a finished answer.

Read it back literally: to get a_2 you need a_1, and the rule does not say what a_1 is. It also describes 5,15,45,... and 100,300,900,... equally well. A recursive description is two statements, and the first term is one of them.

The same gap sinks arithmetic answers: a_n=a_(n-1)+7 alone names no particular sequence either.

How to catch it: Cover the list of terms and try to produce the first one from your rule alone. If you cannot, add the line a_1= … before you move on.