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 𝑟=𝑛(𝑛+1)2.David 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.

  • A1, true
  • B1, false

David 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 David’s assumption and simplify the result to find an expression for 𝑟.

  • A ( 𝑘 + 2 ) 2
  • B ( 𝑘 + 1 ) ( 𝑘 + 2 ) 2
  • C ( 𝑘 + 1 ) ( 𝑘 + 2 )
  • D ( 𝑘 + 2 ) 2

David 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 David’s conclusion correct?

  • AYes
  • BNo

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.

  • A 𝑟 = 𝑟 + ( 𝑘 + 1 ) = 𝑘 ( 𝑘 + 1 ) ( 2 𝑘 + 1 ) 6 + ( 𝑘 + 1 )
  • B 𝑟 = 𝑟 + 1 = 𝑘 ( 𝑘 + 1 ) ( 2 𝑘 + 1 ) 6 + 1
  • C 𝑟 = 𝑟 + ( 𝑟 + 1 ) = 𝑘 ( 𝑘 + 1 ) ( 2 𝑘 + 1 ) 6 + ( 𝑟 + 1 )
  • D 𝑟 = 𝑟 + ( 𝑘 ) = 𝑘 ( 𝑘 + 1 ) ( 2 𝑘 + 1 ) 6 + ( 𝑘 )

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