Proof by induction...sometimes it takes only one geometric series

 We are in the middle of another working week and students to be more comfortable with mathematical proofs. The next exam is coming soon and I reflected on the main overall points that they have to take away from this course. One of them is understanding mathematical induction. 

I started by giving the general idea of induction. 

•  Let's think about a ladder and taking the first step/level and moving to the subsequent levels. 
• We can also think about domino pieces and what happens when the first piece hits the second one and so forth. One of them screamed that he knew what domino was because he was Dominican!  This helps the whole class to wake up 😊

https://www.uptowncollective.com/2019/05/05/dominoes-dominican-chess/

We formalized the idea of mathematical induction and move to the classical example of adding the first n positive integers.

Prove that 1+ 2 + ...+ n= n(n+1)/2

I encouraged them to always do what they need to prove. What is P(n)? 

Is the proposition true for P(1)?  What is the inductive hypothesis? What do I need to prove?

We then discussed the proof of a particular case of the geometric series. I am so glad kept pausing to ask "questions or comments"? Lots of questions about algebra and arithmetic. It made me reflect that many times we make the assumption that students remember the basic properties of powers and real numbers. If we discuss these struggles with the prerequisites, at least we are aware of what needs to be solved and take action.

We don't need to send 20 exercises so they never forget. Many times we just need that particular exercise that makes us reflect on the application of the properties. 

The lecture has a happy ending. The exercise was now proving the general formula of the geometric series by mathematical induction. Many of them were able to prove the basis step, P(0) and they were already starting to apply the inductive step P(k) to prove P(k+1)...we'll continue on Monday. 

Will they be thinking about induction during the weekend? I want to think that at least a seed was planted today.