Common Difference
Functions & Advanced AlgebraThe 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 number you add each time in an arithmetic sequence. It is the same (common) between every pair of consecutive terms.
Example
In $2, 5, 8, 11, 14, \ldots$, the common difference is $3$. In $10, 7, 4, 1, -2, \ldots$, the common difference is $-3$.
Key Insight
You can find the common difference by subtracting any term from the term that follows it. If it is the same every time, the sequence is arithmetic.
Definition
In an arithmetic sequence, the common difference $d = a_{n+1} - a_n$ for all $n$. It can be positive (increasing), negative (decreasing), or zero (constant). The entire sequence is determined by any one term plus $d$.
Example
A sequence starts at $100$ and has $d = -7$: $100, 93, 86, 79, \ldots$ The $15$th term: $a_{15} = 100 + 14(-7) = 100 - 98 = 2$.
Key Insight
The common difference is the slope of the linear function $n \to a_n$. Two sequences with the same $d$ are parallel lines when plotted as functions of $n$.
Definition
The common difference $d$ 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. The Green-Tao theorem (2004) proves that the prime numbers contain arithmetic progressions of arbitrary length.
Example
Arithmetic progressions of $3$ primes: $3, 5, 7$ ($d=2$); $5, 11, 17$ ($d=6$); $7, 19, 31$ ($d=12$). By Green-Tao, such progressions exist of any finite length.
Key Insight
The common difference is the primary structural parameter of an arithmetic progression, analogous to the common ratio for geometric sequences. The two together parameterize the simplest recursive sequences.