Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence.
Formula
S_n = n/2 \cdot (a_1 + a_n)
Definition
An arithmetic series is the sum you get by adding up the terms of an arithmetic sequence, a list of numbers with a constant difference: the sum of the first $n$ terms is $S_n = n/2 \cdot (a_1 + a_n)$, or equivalently $S_n = n/2 \cdot (2a_1 + (n-1)d)$, derived by pairing the first and last terms, the second and second-to-last, and so on. Expanded, $\sum_{k=1}^{n} (a_1 + (k-1)d) = n \cdot a_1 + d \cdot n(n-1)/2$ is a degree-$2$ polynomial in $n$, an instance of the general principle that partial sums of polynomials of degree $k$ are polynomials of degree $k+1$.
Example
Adding $1, 3, 5, 7, 9$ gives the arithmetic series $1 + 3 + 5 + 7 + 9 = 25$. Gauss's famous shortcut sums $1 + 2 + \ldots + 100$ as $S_{100} = 100/2 \cdot (1 + 100) = 5050$, pairing the first and last term ($1+100=101$), the second and second-to-last, and so on, each pair summing to the same total. The sum of the first $n$ squares, $\sum_{k=1}^{n} k^2 = n(n+1)(2n+1)/6$, is a cubic polynomial in $n$, following Faulhaber's formulas, where each increase in the degree of the summand raises the degree of the sum by $1$.
Key Insight
The formula $S_n = n/2 \cdot (a_1 + a_n)$ has an elegant interpretation: the average of the first and last terms multiplied by the number of terms, exactly the area formula for a trapezoid. Faulhaber's formulas, involving Bernoulli numbers, give closed forms for $\sum_{k=1}^{n} k^p$ for any fixed $p$, revealing deep connections between combinatorics, number theory, and the Riemann zeta function.