In this lesson, we will learn how to apply the mathematical induction method to prove a summation formula.

Q1:

Jackson has read in a textbook that 𝑟=𝑛(𝑛+1)2.Jackson wants to prove this using induction.

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

Jackson assumes that the summation formula is true when 𝑛=𝑘 giving him that 𝑟=𝑘(𝑘+1)2. For the induction step, he needs to show that 𝑟=(𝑘+1)(𝑘+2)2. Using the fact that=+(𝑘+1), substitute in Jackson’s assumption and simplify the result to find an expression for 𝑟.

Jackson then makes the following conclusion:

If our assumption is correct for 𝑛=𝑘, we have shown that the summation formula is correct when 𝑛=𝑘+1. Therefore, as we have shown that the summation formula is true when 𝑛=1, by mathematical induction, the formula is true for all natural numbers 𝑛.

Is Jackson’s conclusion correct?

Q2:

Charlotte is trying to prove the summation formula 𝑟=𝑛(𝑛+1)(2𝑛+1)6.

She has checked that the basis is correct, has assumed that 𝑟=𝑘(𝑘+1)(2𝑘+1)6, and is trying to show that 𝑟=(𝑘+1)(𝑘+2)(2𝑘+3)6.

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

Q3:

Natalie wants to prove, using induction, that 𝑓(𝑛)=2−3 is divisible by 5 for all integers 𝑛≥1.

First, she needs to check the base case when 𝑛=1. Substitute 𝑛=1 into the expression and determine the result when it is divided by 5.

Natalie then makes the assumption that 𝑓(𝑘)=2−3 is divisible by 5. She then needs to show that 𝑓(𝑘+1)=2−3() is divisible by 5. To do this, she considers the difference 𝑓(𝑘+1)−𝑓(𝑘). Write this difference in the form 𝑎2−𝑏3.

At this stage it is not clear whether 𝑓(𝑘+1) is divisible by 5. Natalie notices that she may be able to substitute 𝑓(𝑘) into the expression. By writing 72 as 52+22, rewrite the expression for 𝑓(𝑘+1)−𝑓(𝑘) to incorporate 𝑓(𝑘).

Natalie rearranges the equation 𝑓(𝑘+1)=52+3𝑓(𝑘). She then comes to the following conclusion: If the assumption is correct that the expression is divisible by 5 when 𝑛=𝑘, then we have shown that the expression is divisible by 5 when 𝑛=𝑘+1. As we have shown that the expression is divisible by 5 when 𝑛=1, we have proved by mathematical induction that the expression is divisible by 5 for all integers 𝑛≥1.

Is Natalie’s conclusion correct?

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.