Math July 13, 2026 · 8 Min Read

Consecutive Integers Calculator – Guide & Formulas

Find consecutive integers that sum to a target value. Solve for unknown integers in sequences and verify results with step-by-step algebra.

Find consecutive integers that add up to any target sum. Enter how many consecutive integers you need and the target sum to solve for the sequence with algebraic steps.

Key Takeaway

Use the free Consecutive Integers Calculator to find consecutive integers that sum to a target value. solve for unknown integers in sequences and verify results with step-by-step algebra. Get instant results with step-by-step explanations.

How to Use the Consecutive Integers Calculator

  1. Enter the number of consecutive integers in the sequence (e.g., 3 for three consecutive integers).
  2. Enter the target sum you want the integers to add up to.
  3. Review the solution showing the starting integer and the full sequence.
  4. Follow the verification step showing the sum of the integers equals your target.

The Formula

Sum of n consecutive integers starting from x: S = n·x + n(n-1)/2. Solving for x: x = (S - n(n-1)/2) / n.

Variable Definitions

  • S: The target sum of the consecutive integers
  • n: The number of consecutive integers in the sequence
  • x: The first (smallest) integer in the sequence

Three Consecutive Integers That Sum to 72

Find three consecutive integers whose sum is 72.

  1. Step 1: Set up the equation. For 3 consecutive integers starting at x: x + (x+1) + (x+2) = 72.
  2. Step 2: Simplify: 3x + 3 = 72.
  3. Step 3: Solve for x: 3x = 69, so x = 23.
  4. Step 4: The three consecutive integers are 23, 24, and 25.
  5. Step 5: Verify: 23 + 24 + 25 = 72 ✓

Frequently Asked Questions

What are consecutive integers?

Consecutive integers are integers that follow each other in order without gaps. For example, 5, 6, 7 are three consecutive integers. Each integer is exactly 1 more than the previous.

Can consecutive integers be negative?

Yes. Consecutive integers can be any integers: negative, zero, or positive. For example, -3, -2, -1 are consecutive integers, as are 0, 1, 2.

What is the formula for the sum of consecutive integers?

The sum of n consecutive integers starting from x is S = n·x + n(n-1)/2. This simplifies to S = n(x + (n-1)/2).

Can every number be written as a sum of consecutive integers?

No. A number can be written as a sum of 2 or more consecutive integers if and only if it is not a power of 2. Powers of 2 (1, 2, 4, 8, 16, ...) cannot be expressed this way.

How many ways can a number be expressed as consecutive sums?

The number of ways equals the number of odd factors of the number. For example, 15 has odd factors 1, 3, 5, 15, so it can be expressed as a sum of consecutive integers in 4 ways.

What about consecutive even or odd integers?

Consecutive even integers differ by 2 (e.g., 4, 6, 8). Consecutive odd integers also differ by 2 (e.g., 7, 9, 11). The formula changes: for n consecutive even integers starting at 2x, S = 2nx + n(n-1).

How do I find consecutive integers that sum to zero?

Set the sum S = 0 and solve: x = (0 - n(n-1)/2) / n = -(n-1)/2. This only gives integer solutions when n is odd. For example, 3 consecutive integers summing to 0 are -1, 0, 1.

Are there consecutive prime numbers?

The only consecutive prime numbers are 2 and 3. All other consecutive integers differ by 1, and since all integers greater than 2 are odd, one of any pair of consecutive integers must be even (and therefore not prime).