Find consecutive integers that sum to a target value. Perfect for solving algebra word problems and number theory puzzles.
Last updated: 2026-05-24T22:58:31.754Z | By ForgeCalc Engineering
The sum all integers must equal
How many integers to find
Enter a numeric target sum and a positive integer count. The count must be a positive integer.
Yes! Consecutive integers can include negative numbers. For example, -3, -2, -1 are consecutive integers. The formula works the same way regardless of whether numbers are positive or negative.
Consecutive even integers differ by 2 (e.g., 2, 4, 6), as do consecutive odd integers (e.g., 1, 3, 5). For these, use a, a+2, a+4, etc. and adjust the sum formula accordingly.
Not all sums can be achieved with a given number of consecutive integers. When the formula gives a non-integer, no solution exists. For example, no 2 consecutive integers sum to 10.
For an odd number of consecutive integers, the middle one equals Sum ÷ n. If 3 integers sum to 15, the middle is 15 ÷ 3 = 5, giving you 4, 5, 6.
The formula is n(n+1)/2. This was discovered by mathematician Carl Friedrich Gauss, who allegedly found this pattern as a child.
Consecutive integers are a special case of arithmetic sequences where the common difference is 1. The general arithmetic sequence formula applies here with d = 1.
Finding the First Integer
a = (Sum − n(n−1)/2) / n
where a is the first integer, Sum is the target, and n is how many integers
Sum Formula
Sum = n·a + n(n−1)/2
Total when starting at a and adding n consecutive integers
Arithmetic Mean Property
Mean = Sum / n
For odd n, this equals the middle integer
Generating the Sequence
a, a+1, a+2, ..., a+(n−1)
n consecutive integers starting from a