Consecutive Number Sequences — Definition, Formula & Examples
A consecutive number sequence is a list of integers where each number is exactly 1 more than the number before it, such as 5, 6, 7, 8.
A consecutive number sequence is a finite or infinite subset of the integers of the form for some integer and non-negative integer , where each term exceeds its predecessor by exactly 1.
Key Formula
Where:
- = Sum of the consecutive integers
- = First integer in the sequence
- = Number of consecutive integers
How It Works
To build a consecutive number sequence, pick any integer as your starting point and keep adding 1. You can also work with consecutive even numbers (like 2, 4, 6, 8) or consecutive odd numbers (like 7, 9, 11, 13), where each term differs by 2 instead of 1. A useful property is that the sum of consecutive integers starting from equals . Word problems often ask you to find consecutive numbers that satisfy a given condition, like having a particular sum or product.
Worked Example
Problem: Find three consecutive integers whose sum is 72.
Set up variables: Let the three consecutive integers be , , and .
Write the equation: Their sum equals 72.
Solve: Combine like terms and solve for .
Answer: The three consecutive integers are 23, 24, and 25.
Why It Matters
Consecutive number problems appear frequently on standardized tests and in algebra courses. They also build your skill at translating word problems into equations, which is essential throughout high school math and beyond.
Common Mistakes
Mistake: Using , , when the problem asks for consecutive integers (not consecutive even or odd integers).
Correction: Consecutive integers differ by 1, so use , , . Reserve a gap of 2 only when the problem specifies consecutive even or consecutive odd numbers.