Learn Mathematical Induction With A Few Simple Examples

How do Mathematicians prove things?

Photo Credit: Tom Wilson on Unsplash

Recall positive even numbers are 2, 4, 6, 8, … and positive odd numbers are 1, 3, 5, 7,….

Now consider the following mathematical statement:

The sum of the first n positive odd numbers is equal to n²

This is either true or false.

We would only need one example where this doesn’t work to say this statement in general is false.

However, when running the first few values of n, we can see this statement is holding up:

• n=1: 1=1² (the sum of the first 1 odd number is equal to 1²)
• n=2: 1+3=2² (the sum of the first 2 odd numbers is equal to 2²)
• n=3: 1+3+5=3² (the sum of the first 3 odd numbers is equal to 3²)
• n=4: 1+3+5+7=4² (the sum of the first 4 odd numbers is equal to 4²)
• n=5: 1+3+5+7+9=5² (the sum of the first 5 odd numbers is equal to 5²)

How could we prove this is true for all values of n? We cannot check every value of n since there are infinitely many odd numbers.

Mathematicians use a technique called the Principle of Mathematical Induction to solve…