Use a System of Equations To Solve “Number” Problems – Consecutive Integer Partial Sum Example (2011-0021)
“Number – Consecutive Integers “: Use a system of linear equations to find the set of consecutive integers meeting specified conditions.
The sum of the first and last of five consecutive integers is 42. What are the five numbers?
[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. |