Use a System of Equations To Solve “Number” Problems – Consecutive Integer Partial Sum Example (2011-0021)

Problem Type:

“Number – Consecutive Integers “: Use a system of linear equations to find the set of consecutive integers meeting specified conditions.

Problem:

The sum of the first and last of five consecutive integers is 42. What are the five numbers?

Solution

[I plan to provide a more detailed solution in the near future. In the meantime I hope you find the following abbreviated solution helpful.]

By definition, the set of integers is made up of the counting numbers {1, 2, 3, 4, …}, zero {0}, and the opposite of the counting numbers {-1, -2, -3, -4, …}. According to merriam-webster.com consecutive means “following one after the other in order.” Therefore, consecutive integers are integers that follow one right after the other, for example: {-2, -1, 0, 1, 2, 3}.

If we let N represent the first of the five consecutive integers, then (N + 1) represents the second, (N + 2) represents the third, (N + 3) represents the fourth, and (N + 4) represents the fourth.

We now have everything we need to write an algebraic equation that can be used to find N, the first integer. I recommend that you add the expressions vertically, as shown below, to generate the equation for the sum.

♦ At this point we need to be very careful. Remember, we are not told the sum of the five — only that “The sum of the first and last of five consecutive integers is 42.” Using this information, along the appropriate expressions from above, we get …

♦ To solve for N we need to (1) “undo” adding 4 by subtracting 4 from both sides of the equation, then (2) “undo” multiplying by 2 by dividing both sides of the equation by 2.

♦ To find the integers substitute 19 for N in your original expressions.