Math 1 · Unit 9 · Lesson 3

Geometric Sequences, Exponentials Made of Dots

A geometric sequence multiplies by the same number each time, the common ratio r. Find r by dividing a term by the one before. Bigger than 1, the terms grow; between 0 and 1, they shrink. The recursive rule multiplies the term before by r, and the explicit rule is a sub 1 times r to the n minus 1, because the first term has not been multiplied yet. The common ratio is the growth factor from Unit 7, and the terms are dots on the exponential curve.

About 45 minutes The common ratio is the growth factor

The big idea

Multiplying gives geometric. A geometric sequence multiplies by the same number, the common ratio r, to get each next term, and you find r by dividing any term by the one before it. If r is bigger than 1 the terms grow, and if r is between 0 and 1 they shrink, the same growth and decay rule as Unit 7. The recursive rule is a sub 1 for the start and a sub n equals r times a sub n minus 1. The explicit rule is a sub n equals a sub 1 times r to the n minus 1, with n minus 1 because the first term has not been multiplied yet. The common ratio is the growth factor b, and the terms are dots on the exponential curve. When the ratio is a fraction, the terms shrink toward 0 but never reach it, and a value like f(3) = 50 is read in the story's words: after 3 halvings, 50 is left.

The parts of this lesson

Each part is its own short page. Go in order, or pick the one you need.

1The Words We UseThe vocabulary for this lesson, in one place.Start here 2Multiplying by the same number: the common ratioAdding gives arithmetic, multiplying gives geometric. 3It is an exponential: the ratio is the growth factorThe dots sit on an exponential curve, so a geometric sequence is the exponential family from Unit 7 in... 4A fraction ratio, and what f(3) = 50 meansA decay example, written in function notation. 5Practice ProblemsTry them, get help one step at a time, then print how it went.

Printables

One page with the common ratio, both rules, the n minus 1 reason, the link to the growth factor, a fraction ratio, and what a function value means. Plus a worksheet with an answer key and a short lesson plan.

Study guide (PDF) Worksheet and answer key (PDF) Lesson plan (PDF)

Where this goes next

Previous lesson: Arithmetic Sequences, Lines Made of DotsThe common difference is the slope. Next lesson: Recursive and Explicit, Both WaysChange a recursive rule into an explicit one, and back. Back to the Math 1 mapEvery Math 1 lesson, in course order. Tell us what you thoughtA few quick questions about this lesson. It takes a minute, and it is how the lesson gets better.

NC Math 1 standards in this lesson: NC.M1.F-IF.3 (terms of a geometric sequence come from an exponential function), NC.M1.F-BF.1a (build exponential functions, including geometric sequences), NC.M1.F-BF.2 (explicit and recursive forms), NC.M1.F-IF.2 (interpret function notation in context).