Arithmetic Pattern
An arithmetic pattern is a sequence in which each term is obtained from the previous one by adding or subtracting a fixed constant, called the common difference.
Formula
a_n = a_1 + (n - 1)d
Definition
An arithmetic pattern is a number sequence where you always add (or subtract) the same amount, called the common difference, to get the next term; think of it like skip-counting by the same number over and over (by 2s, 5s, 10s). An arithmetic sequence has a constant common difference $d$ between consecutive terms, with $n$th term formula $a_n = a_1 + (n-1)d$, where $a_1$ is the first term, and the sum of the first $n$ terms is $S_n = n(a_1 + a_n)/2$ (Gauss's sum formula). Formally, an arithmetic sequence satisfies the recurrence $a_{n+1} - a_n = d$; its closed form is linear in $n$, its partial sum $S_n = na_1 + dn(n-1)/2$ is quadratic in $n$, and it is a first-order linear recurrence with constant coefficients, solvable by the characteristic equation method.
Example
$5, 10, 15, 20, 25$ is an arithmetic pattern with common difference $5$. Sequence $7, 11, 15, 19, \ldots$ has first term $a_1 = 7$ and common difference $d = 4$; the $10$th term is $a_{10} = 7 + 9(4) = 43$. The sum $1 + 2 + \ldots + 100 = 100 \cdot 101/2 = 5050$ (Gauss) is an arithmetic series with $a_1 = 1$, $d = 1$, $n = 100$, and in general $S_n = n(n+1)/2$ defines the triangular numbers.
Key Insight
Arithmetic sequences correspond to linear functions, $a_n = dn + (a_1 - d)$, so graphing $(n, a_n)$ gives a straight line, connecting sequences to linear algebra. Gauss reportedly computed $1 + 2 + \ldots + 100$ as a child by pairing first and last terms, $(1+100) + (2+99) + \ldots$, giving $50$ pairs of $101 = 5050$, an insight that generalizes to the closed-form sum formula for any arithmetic series.