How to Solve Consecutive Integer Problems: SAT, ACT, & Other Exams
Consecutive integer math problems pose a unique challenge because you must solve for a set of integers with only one variable. Standardized tests such as the SAT, ACT, GRE, GMAT, and other assessment and entrance exams frequently use consecutive integer problems to test your critical thinking and algebra skills.
Luckily, these problems are easy to solve once you know the right technique. You can also use the online solver.
- Represent the smallest integer by n.
- Call the successive integers n+1, n+2, etc. for as many integers are given in the problem.
- Plug the variables n, n+1, n+2, etc. into the problem and solve for n, the smallest integer.
- Use this value to solve for n+1, n+2, etc. to find the complete set.
- Use these values to perform any additional calculations the problem requires
Variation: If the questions deals with consecutive even integers or consecutive odd integers, denote the smallest integer by n. Then, denote the successive integers by n+2, n+4, n+6, etc. Since the difference between any two consecutive even or odd integers is 2, the sequence jumps by 2 rather than by 1.
Example 1: The sum of the squares of four consecutive positive integers is 174. What is the sum of these integers?
First, denote the integers in the set by n, n+1, n+2, and n+3. Since we are given information about the sum of their squares, we set up the equation
n² + (n+1)² + (n+2)² + (n+3)² = 174.
This equation can be simplified and solved as follows
n² + (n+1)² + (n+2)² + (n+3)² = 174
4n² + 12n + 14 = 174
4n² + 12n - 160 = 0
n² + 3n - 40 = 0
n = 5, n = -8
Since the problem states "positive integers," the smallest integer must be 5. This means the set of four consecutive integers is 5, 6, 7, and 8. Since 5 + 6 + 7 + 8 = 26, the answer is 26.
Example 2: The product of three consecutive even integers is 4032. What is the middle integer of this set?
Call the three integers n, n+2, and n+4. Then set up the equation
n(n+2)(n+4) = 4032
n³ + 6n² + 8n = 4032
n³ + 6n² + 8n - 4032 = 0
You can solve this equation with a cubic formula calculator or an exam-approved graphing calculator. The solution is n = 14. This means the set of consecutive even integers is 14, 16, and 18. The middle number is 16.
Example 3: In a set of five consecutive odd integers, the product of the two smallest integers is 2 more than the sum of the remaining integers. What is the largest number in the set?
The five consecutive odd integers can be represented by n, n+2, n+4, n+6, and n+8. The equation that relates the quantities is
n(n+2) = (n+4) + (n+6) + (n+8) + 2.
This can be simplified and solved as follows:
n² + 2n = 3n + 20
n² - n - 20 = 0
n = 5, n = -4
Since n must be odd, the smallest number in the set must be 5. This means the set is 5, 7, 9, 11, and 13. The largest among these is 13.
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