Slides for 5/10 & 5/11 Precalculus

Warm-Up Set 13 Problem 3 Write an explicit formula that generates this sequence: 60, 57, 54, 51, 48, 45, . . .

Objectives Today, we will: Identify geometric sequences and the common ratio of a geometric sequence Write formulas for geometric sequences Find the partial sum of a geometric sequence Solve problems using geometric sequences and series Compare and contrast arithmetic sequences and geometric sequences

Geometric Sequences A geometric sequence has a common ratio between each pair of terms – we multiply or divide by the same value in each step. The recursive formula for a geometric sequence is , where r is the common ratio. The explicit formula for a geometric sequence is . Note that this is essentially an exponential equation!

Finding a Geometric Series We use these formulas to find the partial sum up to the nth term for a geometric sequence. Use the first equation when r > 1, and the second when r < 1. They’re mathematically equivalent.

Sample Problem A rubber ball is dropped from a height of 2 meters onto the floor. It rebounds to 90% of its previous height on each bounce. How high does it rebound on the 4th bounce? How far does it travel on each bounce? What is the total distance it has traveled between the first time it hits the floor and the 6th time it hits the floor? (Count carefully!)

Are All Sequences Either Arithmetic Or Geometric? No.

Aww. The good news is, most of the sequences you will encounter will be one of the two. The other common ones are binomial sequences, which we will talk about later in this unit in more detail, and quadratic ones, like square and triangular numbers.

So How Do We Tell Them Apart? If there is a common difference between terms, then it’s an arithmetic sequence. If there’s a common ratio between terms, then it’s a geometric sequence. If it doesn’t have either a common difference or a common ratio, it must be something else! If the difference of the differences (the second difference) is common to all the terms, then it’s a quadratic of some sort. Otherwise, look for other patterns.