Most mathematical proofs will require some explanation of why the proof works. Some text is added to the proof in the following example to help with the explanation. Try to follow why this proof works.


Example 3

Consider the following conjecture:

  • The average of three consecutive Integers is equal to the middle integer.
  1. Give two examples showing the conjecture works.

    The average of 3, 4, 5 is .
    The average of −8, −7, −6 is .


  1. Prove the conjecture.

    In order to show that the conjecture is always true, you can use variables to show that the conjecture will work for any number. This can be done by showing that the average can be represented in exactly the same way as the middle Integer.

    Let x represent the smallest Integer.
    The three Integers can then be represented as x , x + 1, and x + 2.

    The average of these three values is .

    Simplify.



    So the average can be represented by x + 1, which is also the middle number from the original list of numbers, x, x + 1, and x + 2. It has been shown that the average of three consecutive Integers is equal to the middle Integer and so the proof is complete.

Sometimes the results of one proof can be used in a different proof. Once you have successfully proven a particular relationship to be true, you can use that relationship as fact in proofs moving forward. Example 4 will prove a relationship that will be used in the proof shown in Example 5.