The main purpose of today’s lesson is to formalize the work we have done over the last two days and to work out the nuts and bolts of describing sequences and series. I want students to be able to describe a sequence recursively and explicitly and to be able to express a series using summation notation.
The format for today’s lesson will be a whole class discussion with breaks for students to work with their table. I begin by going over slide 2 of the PowerPoint and see if students recall how to find the explicit and recursive formula for the arithmetic sequence that is given. Usually a few students remember the notation of an, but if my class has not learned it yet, I show them.
One common misconception for the recursive formula is that students only have to say that an = 3 + an-1, so I will ask them what the recursive formula for 1, 4, 7, 10, 13, … would be. Since it is a completely different sequence, we need to use a different recursive formula, so students will see the need for identifying the first term.
After going through an example, I show slide 3 and give students about 5 minutes to work on that with their table. This time we have a geometric sequence and I want them to write the explicit and recursive formula. Additionally, they will see if they remember the format for summation notation. After they work on this, I will randomly select students to share their work. For the summation notation, students may remember bits and pieces of it, but usually in the whole class discussion we can get it correct. I also introduce the term “partial sum” (in this case we found the fifth partial sum) and its notation (S5). Then I take some time to show students how to find the sum on their graphing calculator.
Finally, the last slide has the Fibonacci sequence and asks students to describe it using a recursive formula. This will be difficult for students since it is not arithmetic or geometric. I give them about 5-7 minutes to work on this. Since there is not a constant change from one term to the next, students will have to get creative in their approach. Having them describe it in words can be helpful. If they notice that each term is the sum of the previous two terms, they might be able to write it as an = an-1 + an-2. They may then look at the other two recursive formulas and just define the first term. When I go over this with the class, I will start with just defining the first term and actually plugging in 2 for n and showing that we do not have enough information to figure out what the term should be.
After our class discussion, students have time to begin the homework assignment. This assignment summarizes the main points of sequences and series and allows students to get some practice with using recursive and explicit formulas and summation notation.
Teacher Note: I have included some problems from the introductory section of sequences and series from our class textbook on the worksheet. You will certainly want to change those based on your class textbook (unless you use the same one as me).
Here is a video where I discuss today’s homework assignment.