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