Consecutive Numbers
ArithmeticConsecutive 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 no numbers skipped.
Example
$7, 8, 9, 10$ are four consecutive numbers. $2, 4, 6$ are consecutive even numbers (even numbers in order).
Key Insight
Consecutive numbers differ by exactly $1$. Consecutive even numbers differ by $2$. Consecutive odd numbers also differ by $2$.
Definition
Consecutive integers are of the form $n, n+1, n+2, \ldots, n+k$. Consecutive even integers: $n, n+2, n+4, \ldots$ ($n$ even). Consecutive odd integers: $n, n+2, n+4, \ldots$ ($n$ odd). The sum of $n$ consecutive integers starting at $a$ is $na + n(n-1)/2$.
Example
Three consecutive integers summing to $48$: $n + (n+1) + (n+2) = 48$, so $3n + 3 = 48$, $n = 15$. The integers are $15, 16, 17$. Check: $15+16+17 = 48$.
Key Insight
The average of any set of consecutive integers equals the middle value (or the average of the two middle values). This makes summing consecutive integers easy: $\text{sum} = \text{count} \times \text{average}$.
Definition
Consecutive integer problems connect to arithmetic progressions: $n, n+1, \ldots, n+k$ is an AP with first term $n$ and common difference $1$. The product of $k$ consecutive integers is always divisible by $k!$ (combinatorial identity: $\binom{n+k}{k} = \frac{(n+1)(n+2)\cdots(n+k)}{k!}$ is an integer). This is the basis for proving that binomial coefficients are integers.
Example
Proof that 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 products (falling factorials) satisfy identities analogous to powers in calculus. The difference operator on consecutive products mirrors the derivative for continuous functions, unifying discrete and continuous mathematics.