Arithmetic Pattern
Pre-AlgebraAn 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 to get the next term. That amount is called the common difference.
Example
$5, 10, 15, 20, 25$ is an arithmetic pattern. You add $5$ each time. The common difference is $5$.
Key Insight
Think of an arithmetic pattern like counting by the same number over and over. Skip-counting (by 2s, 5s, 10s) creates arithmetic patterns.
Definition
An arithmetic sequence (pattern) has a constant common difference $d$ between consecutive terms. The $n$th term formula is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term. The sum of the first $n$ terms is $S_n = n(a_1 + a_n)/2$ (Gauss's sum formula).
Example
Sequence: $7, 11, 15, 19, \ldots$ First term $a_1 = 7$, common difference $d = 4$. The $10$th term: $a_{10} = 7 + 9(4) = 7 + 36 = 43$.
Key Insight
Arithmetic sequences correspond to linear functions: $a_n = dn + (a_1 - d)$, which is linear in $n$. Graphing $(n, a_n)$ gives a straight line, connecting sequences to linear algebra.
Definition
An arithmetic sequence is a sequence satisfying the recurrence $a_{n+1} - a_n = d$ (constant). Its closed form $a_n = a_1 + (n-1)d$ is a linear function of $n$. The sum $S_n = na_1 + dn(n-1)/2$ is quadratic in $n$. Arithmetic sequences are first-order linear recurrences with constant coefficients, solvable by the characteristic equation method.
Example
The sum $1 + 2 + \ldots + 100 = 100 \cdot 101/2 = 5050$ (Gauss). This is an arithmetic series with $a_1 = 1$, $d = 1$, $n = 100$. In general, the partial sum $S_n = n(n+1)/2$ defines the triangular numbers.
Key Insight
Gauss reportedly computed $1 + 2 + \ldots + 100$ as a child by pairing first and last terms: $(1+100) + (2+99) + \ldots = 50$ pairs of $101 = 5050$. This insight generalizes to the closed-form sum formula for any arithmetic series.