Consecutive Numbers
Consecutive numbers are integers that follow each other in order without any gaps, such as 4, 5, 6, 7.
Formula
n, n+1, n+2, \ldots
Definition
Consecutive numbers are numbers that come one after another in counting order, with none skipped; consecutive integers take the form $n, n+1, n+2, \ldots, n+k$, differing by exactly $1$, while consecutive even or odd integers differ by $2$. The sum of $n$ consecutive integers starting at $a$ is $na + n(n-1)/2$, and the average of any set of consecutive integers equals the middle value (or the average of the two middle values), which makes summing them easy. More generally, consecutive-integer problems connect to arithmetic progressions, and the product of $k$ consecutive integers is always divisible by $k!$, a combinatorial identity ($\binom{n+k}{k} = \frac{(n+1)(n+2)\cdots(n+k)}{k!}$ is an integer) that underlies the proof that binomial coefficients are integers.
Example
$7, 8, 9, 10$ are four consecutive numbers, and $2, 4, 6$ are consecutive even numbers. Three consecutive integers summing to $48$: $n + (n+1) + (n+2) = 48$, so $3n + 3 = 48$ and $n = 15$, giving $15, 16, 17$ (check: $15+16+17=48$). The product of any $3$ consecutive integers is divisible by $6$: one of them is divisible by $3$ and at least one by $2$, more precisely $n(n+1)(n+2) = 6\binom{n+2}{3}$, which is $6$ times an integer.
Key Insight
Consecutive numbers differ by exactly $1$; consecutive even or odd numbers differ by $2$. Consecutive products (falling factorials) satisfy identities analogous to powers in calculus, and the difference operator on consecutive products mirrors the derivative for continuous functions, unifying discrete and continuous mathematics.