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Consecutive Numbers — Definition, Formula & Examples

Consecutive numbers are numbers that follow each other in order, each one exactly 1 more than the previous. For example, 5, 6, 7 are three consecutive numbers.

A set of consecutive integers is a finite sequence n,n+1,n+2,…,n+kn, n+1, n+2, \ldots, n+k where nn is any integer and kk is a non-negative integer. The term extends to consecutive even numbers (differing by 2), consecutive odd numbers (differing by 2), and other arithmetic patterns.

Key Formula

n,  n+1,  n+2,  …,  n+kn,\; n+1,\; n+2,\; \ldots,\; n+k
Where:
  • nn = The first integer in the sequence
  • kk = The number of steps after the first integer (so there are k + 1 total numbers)

How It Works

To represent consecutive numbers in algebra, pick a variable for the first number and add 1 for each number after it. If the first number is nn, then three consecutive numbers are nn, n+1n+1, and n+2n+2. For consecutive even numbers, use nn, n+2n+2, n+4n+4 (where nn is even). Consecutive odd numbers follow the same spacing: nn, n+2n+2, n+4n+4 (where nn is odd). This technique lets you translate word problems into equations you can solve.

Worked Example

Problem: The sum of three consecutive numbers is 72. Find the numbers.
Set up variables: Let the three consecutive numbers be nn, n+1n+1, and n+2n+2.
Write the equation: Their sum equals 72.
n+(n+1)+(n+2)=72n + (n+1) + (n+2) = 72
Solve: Combine like terms and solve for nn.
3n+3=72  ⟹  3n=69  ⟹  n=233n + 3 = 72 \implies 3n = 69 \implies n = 23
Answer: The three consecutive numbers are 23, 24, and 25.

Why It Matters

Consecutive number problems appear frequently on standardized tests and in algebra courses. They also show up in number theory proofs — for instance, the product of any two consecutive integers is always even, a fact used in combinatorics and divisibility arguments.

Common Mistakes

Mistake: Using nn, n+2n+2, n+4n+4 for consecutive numbers instead of consecutive even or odd numbers.
Correction: Consecutive numbers differ by 1, so use nn, n+1n+1, n+2n+2. Reserve the spacing of 2 for consecutive even or consecutive odd numbers specifically.