Consecutive Integers Calculator - Sequences, Sums & Analysis

Consecutive integers are whole numbers that follow one after another with no gaps — for example 7, 8, 9, 10. They appear in countless word problems in algebra, in identities of number theory (the sum of the first n positive integers, Gauss's famous schoolboy formula), in combinatorics (choosing k consecutive items from a row), and in everyday situations such as scheduling, page numbering, and labelling. This calculator supports three closely related problems that cover almost every practical question you might ask about a run of consecutive integers. The Generate mode takes a starting integer s and a count n and returns the sequence s, s+1, …, s+n−1, together with its sum and average. The sum follows directly from the arithmetic-series formula: sum = n·(2s + n − 1) / 2. The average is simply s + (n − 1)/2, which is also the midpoint of the sequence. Use this mode when you know where you want to start and how many terms you need. The Find by Target Sum mode reverses the question. Given a target sum T and a count n, the starting integer is s = T/n − (n − 1)/2, which is the average of the sequence shifted back to the first term. The sequence exists with integer terms if and only if s is itself a whole number; this happens exactly when 2T is divisible by n and T/n − (n − 1)/2 is an integer. Use this mode when you have a target — for instance, you want three consecutive integers that add up to 33 — and you need to find the sequence that produces it. The Analyze mode accepts a comma- or space-separated list of integers and checks whether they form a consecutive sequence. It does so by sorting them, computing successive differences, and confirming that every difference equals 1. It also reports the sum, the average, and the count, regardless of whether the sequence is consecutive. Use this mode to validate data, to check a student's answer, or to summarise a list of numbers that you suspect are sequential. A few subtleties are worth noting. Consecutive integers are not the same as consecutive even or consecutive odd integers, which step by two instead of one. The sum of n consecutive integers starting from s is always equal to n times the middle term (or the average of the two middle terms when n is even); this gives an elegant manual shortcut that avoids the full formula. Finally, although the calculator allows negative starting values, every property above still holds without modification, since the arithmetic-series identities are symmetric under sign changes.