Worksheet: Proving Divisibility by Mathematical Induction
In this worksheet, we will practice using the mathematical induction method to prove divisibility by finding a counterexample that disproves a given mathematical statement.
Yasmine wants to prove, using induction, that is divisible by 5 for all integers .
First, she needs to check the base case when . Substitute into the expression and determine the result when it is divided by 5.
Yasmine then makes the assumption that is divisible by 5. She then needs to show that is divisible by 5. To do this, she considers the difference . Write this difference in the form .
At this stage it is not clear whether is divisible by 5. Yasmine notices that she may be able to substitute into the expression. By writing as , rewrite the expression for to incorporate .
Yasmine rearranges the equation . She then comes to the following conclusion: If the assumption is correct that the expression is divisible by 5 when , then we have shown that the expression is divisible by 5 when . As we have shown that the expression is divisible by 5 when , we have proved by mathematical induction that the expression is divisible by 5 for all integers .
Is Yasmine’s conclusion correct?