Write an explicit rule for geometric sequences using the common ratio and any term in the sequence.
Apply understanding of how geometric sequences grow and knowledge of exponents and roots to determine the common ratio and find missing terms.
Compare arithmetic and geometric sequences.
Quick Lesson Plan
Little Red Riding Hood is on her way to grandma’s house but something must be wrong with her pockets, because she keeps dropping crumbs of bread--and in a rather unusual pattern! In this lesson, students learn about (or review) geometric sequences by describing the number of crumbs Little Red drops on successive tiles. First, we want students to see the repeated multiplication so we have them write the number of crumbs using 2s and 3s only. Though some students may be ready for exponential reasoning right away, we find that writing it out long-hand reinforces important algebra concepts that students may forget along the way (when do I add exponents and when do I multiply??). This also leads students smoothly to question 4 where they have to write an explicit rule for the sequence. As you are monitoring students, listen for students that are able to articulate why there is an (n-1) in the exponent.
In question 5, we return to yesterday's idea that any term of a sequence can be an “anchor point” for determining other terms. Knowing the number of crumbs dropped on the 13th tile is sufficient for determining the number of crumbs dropped on the 15th tile because compared to the 13th tile, the number of crumbs on the 15th tile will be 3x3 or 9 times greater. This reasoning may seem obvious to us, and should feel intuitive to students, but we find that students often resort to using formulas and procedures that are actually less efficient, instead of making use of structure to solve a problem.
What does the 2 represent in your table? What does the 3 represent?
How is the number of crumbs changing? Is this an arithmetic sequence?
If I only told you how many she dropped on the 13th tile and the fact that the number of crumbs triples, could you figure out how many she dropped on the 10th tile? The 1st tile? How would you do it?
What’s the average of the numbers 2, 4, 6, 8, and 10? What’s the average of the numbers 2, 4, 8, 16, 32? Is the strategy for both the same? Why or why not?
The focus of the debrief is to look at patterns exhibiting repeated multiplication and to differentiate this from patterns demonstrating repeated addition. We also want students to generalize the explicit formula for the nth term of a geometric sequence and to realize there are multiple ways of doing this.
As mentioned in yesterday’s lesson, we use the idea of a partial sum to differentiate between growth in arithmetic and geometric sequences. While the terms of an arithmetic sequence are evenly distributed, the terms of a geometric sequence are not. For increasing geometric sequences, the increase in consecutive terms gets successively larger. For decreasing geometric sequences, the decrease in consecutive terms gets successively smaller. Note that we do not actually introduce or explore the partial sum formula for geometric sequences. This is one notable way AP Precalculus may differ from your regular Precalculus course.