Consecutive integers

Basic characteristics

Key properties that often appear in elementary number theory and algebra include:

• Equal mean and median: In any finite list of consecutive integers, the arithmetic mean equals the median because the distribution is symmetric; see a standard discussion of mean and median here.
• Arithmetic progression: Consecutive integers have common difference 1, so formulas for arithmetic sequences apply.
• Parity alternation: Consecutive integers alternate between even and odd, so any two consecutive integers are relatively prime.
• Sets and ranges: A contiguous integer set {a, a+1, ..., b} contains b-a+1 elements; more on finite sets appears in related material.

Formulas and examples

Sums and averages of consecutive integers have simple closed forms. The sum of k consecutive integers starting at n is k*n + k(k-1)/2, which follows from summing an arithmetic series. If the list is symmetric around zero, such as -m,...,0,...,m, the sum is zero. Small examples help: the three consecutive integers 4, 5, 6 have sum 15 and mean 5; negatives like -3, -2, -1 sum to -6 and mean -2.

Uses and problem solving

Consecutive integers are frequently used in algebraic problem solving and proofs by contradiction or parity arguments. Typical contest problems ask for the existence of consecutive integers with a given property (e.g., consecutive integers whose product has a specified remainder). Their alternating parity and simple divisibility traits make them useful in modular arithmetic and combinatorics.

Distinctions and related concepts

Closely related ideas include consecutive even integers (numbers like 2n, 2n+2, ... with difference 2) and consecutive odd integers (2n+1, 2n+3, ...). Consecutive integers should not be confused with consecutive powers or consecutive primes; consecutive primes are primes that appear consecutively in the ordered list of primes, and they do not follow a fixed additive step. For background reading and examples, consult an elementary number theory source here.