Common Difference
The common difference is the constant value added to each term in an arithmetic sequence to obtain the next term.
Formula
d = a_{n+1} - a_n
Definition
The common difference is the constant number you add each time in an arithmetic sequence, the same value between every pair of consecutive terms: $d = a_{n+1} - a_n$ for all $n$. It can be positive (increasing), negative (decreasing), or zero (constant), and the entire sequence is determined by any one term plus $d$. Algebraically, $d$ is the generator of the arithmetic progression as a coset of $d\mathbb{Z}$ in $(\mathbb{Z}, +)$; in finite fields $\mathbb{F}_p$, arithmetic progressions of length $k$ are studied in additive combinatorics, and the Green-Tao theorem (2004) proves that the prime numbers contain arithmetic progressions of arbitrary length.
Example
In $2, 5, 8, 11, 14, \ldots$, the common difference is $3$, and in $10, 7, 4, 1, -2, \ldots$, it is $-3$. A sequence starting at $100$ with $d = -7$ gives $100, 93, 86, \ldots$, and its $15$th term is $a_{15} = 100 + 14(-7) = 2$. Arithmetic progressions of three primes include $3, 5, 7$ ($d=2$), $5, 11, 17$ ($d=6$), and $7, 19, 31$ ($d=12$), and by the Green-Tao theorem such progressions exist of any finite length.
Key Insight
You can find the common difference by subtracting any term from the one that follows it; if it is the same every time, the sequence is arithmetic, and geometrically $d$ is the slope of the linear function $n \to a_n$, so two sequences with the same $d$ are parallel lines when plotted against $n$. The common difference is the primary structural parameter of an arithmetic progression, analogous to the common ratio for geometric sequences, and the two together parameterize the simplest recursive sequences.