# Worksheet: Mathematical Induction

In this worksheet, we will practice applying the mathematical induction method to prove a summation formula.

**Q1: **

David has read in a textbook that David wants to prove this using induction.

First, he starts with the basis step substituting into each side of the equation. He calculates that the left-hand side, equals 1. Calculate the value of the right-hand side, and, hence, determine if the basis is true.

- A1, true
- B1, false

David assumes that the summation formula is true when giving him that For the induction step, he needs to show that Using the fact that substitute in David’s assumption and simplify the result to find an expression for

- A
- B
- C
- D

David then makes the following conclusion:

If our assumption is correct for , we have shown that the summation formula is correct when . Therefore, as we have shown that the summation formula is true when , by mathematical induction, the formula is true for all natural numbers .

Is David’s conclusion correct?

- AYes
- BNo

**Q2: **

Charlotte is trying to prove the summation formula

She has checked that the basis is correct, has assumed that and is trying to show that

Charlotte knows that she needs to express in terms of her assumption for the , but she cannot quite remember the method. Determine which of the following is correct.

- A
- B
- C
- D

**Q3: **

Natalie 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.

Natalie 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 .

- A
- B
- C
- D
- E

At this stage it is not clear whether is divisible by 5. Natalie notices that she may be able to substitute into the expression. By writing as , rewrite the expression for to incorporate .

- A
- B
- C
- D
- E

Natalie 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 Natalie’s conclusion correct?

- AYes
- BNo