SWBAT use mathematical induction to prove the formula for the nth partial sum of a sequence.

What happens when a sequence is neither arithmetic nor geometric - can we still find the nth partial sum?

15 minutes

Up to this point we have used formulas to find the nth partial sum of a geometric or arithmetic series. Today we are going to look at what will happen when a sequence is neither and think about how we can still prove that the our formula will work. The main focus of today is to use proof by mathematical induction to **transition from inductive reasoning to deductive reasoning**.

I begin by giving students this worksheet and have them answer questions #1-3 with their table groups for about six minutes. This will get students thinking inductively to** find a pattern for the nth partial sum** of the given sequence. My students will usually notice right away that the nth partial sum is n/(n + 1). Those that did not find this pattern were usually not writing their sums as fractions in lowest terms. You might suggest that if a student is stuck.

After students have time to finish, I will randomly call on students to give their answers for these questions. When I ask what the sum of the first 100 terms is, I will make a big deal about how shocked I am that they came to that answer so quickly. Once they explain that they just found a pattern, it is the perfect segue into question #2 when I ask them what type of reasoning they just used. They may need a reminder that it is called inductive reasoning and it is **not a valid form a proof **– just because it works for a few examples does not mean it will always work.

Next I say that we have to use a valid form of proof, and I ask them for some types of proofs that they already know. Usually I will get the following examples:

- 2-column
- paragraph
- algebraic
- coordinate
- proof by contradiction

20 minutes

After reviewing some types of proofs that students have already used, I say that we are going to learn a **new form of proof called mathematical induction**. I give students the three steps of the process:

- Prove that the formula works for
*n*= 1 - Assume that the formula is true for
*n = k* - Prove that the formula must then be true for
*n = k*+ 1

I use an analogy of a chain of dominoes to explain the process. I talk more about this in the video below.

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Once students understand the general procedure, I will walk them through this example. I will randomly call on students to answer questions throughout the process to make sure that they understand as we go. In this video I go through my strategy for getting from step 2 of induction to step 3 by **writing what their goal is**.

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After this proof has been completed, I **go back to the domino analogy and revisit the concept one more time**. This process is very abstract, so we again talk about how we proved that the formula works for *n* = 1, and then we proved that it will work for *k* + 1, so therefore the formula has to work for *n* = 1 + 1, or *n* = 2. Then we know it will work for the next value, and the next value, and so on.

20 minutes

Next, students will work through #6 from the worksheet with their table groups to see if they can replicate the process. I will usually set up the goal together with the entire class so they know what they are shooting for. After they have time to complete it, I will choose a student to share their work on the document camera.

I stress that they are **writing a valid argument that needs to be clear and easy to follow**. They must be explicit in their steps and explain their thinking – just like if they were making an assertion in an English paper. If in an English paper they made the claim that Brutus was not an honorable man – they would give evidence to support their claim. Proving in math is no different; if you claim that two expressions are equal, you should have evidence to support that.

Finally, I will assign 3 – 4 questions from their textbook to give them some practice with mathematical induction.