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.

First integer:    N
Second integer:   N + 1
Third integer:   N + 2
Fourth integer:   N + 3
Fifth integer:   N + 4

♦ 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 …

First integer:    N
Fifth integer:   N + 4
Sum of first & last: 2N + 4 = 42

♦ 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.

(1) Subtract 4 from both sides of the equation & simplify:
  2N + 4 – 4 = 42 – 4
  2N = 38
(2) Divide both sides of the equation by 2 and simplify:
  2N ÷ 2 = 38 ÷ 2
 

N = 19

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

First integer:    N = 19
Second integer:   N + 1 = 19 + 1 = 20
Third integer:   N + 2 = 19 + 2 = 21
Fourth integer:   N + 3 = 19 + 3 = 22
Fifth integer:   N + 4 = 19 + 4 = 23
Check: Verify that all conditions of the problem are satisfied.
(1) [The numbers are consecutive.]
(2) [The sum of the first and last of five consecutive integers is 42.]
  19 + 23 =? 42
  42 = 42 √
All conditions are satisfied; thus, the five consecutive integers are 19, 20, 21, 22, and 23.