Arithmetic Series
Functions & Advanced AlgebraAn 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 total you get by adding up the terms of an arithmetic sequence. You are summing a list of numbers with a constant difference.
Example
Add the sequence $1, 3, 5, 7, 9$: the arithmetic series is $1 + 3 + 5 + 7 + 9 = 25$. You are summing five consecutive odd numbers.
Key Insight
A famous shortcut: pair the first and last term ($1 + 9 = 10$), then the second and second-to-last ($3 + 7 = 10$). You get pairs of equal sums. This was Gauss's trick for adding $1 + 2 + \ldots + 100$.
Definition
The sum of the first $n$ terms of an arithmetic sequence is $S_n = n/2 \cdot (a_1 + a_n)$, or equivalently $S_n = n/2 \cdot (2a_1 + (n-1)d)$. This is derived by pairing the first and last terms, second and second-to-last, and so on.
Example
Sum $1 + 2 + 3 + \ldots + 100$: $S_{100} = 100/2 \cdot (1 + 100) = 50 \cdot 101 = 5050$. This is Gauss's answer. Using the formula directly saves summing $100$ numbers.
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. This is exactly the area formula for a trapezoid.
Definition
The arithmetic series $\sum_{k=1}^{n} (a_1 + (k-1)d) = n \cdot a_1 + d \cdot n(n-1)/2$ is a degree-$2$ polynomial in $n$. This fact means that partial sums of arithmetic sequences are quadratic, an instance of the general principle that partial sums of polynomials of degree $k$ are polynomials of degree $k+1$.
Example
The sum of the first $n$ squares: $\sum_{k=1}^{n} k^2 = n(n+1)(2n+1)/6$, a cubic polynomial in $n$. Each "level up" in the degree of the summand increases the degree of the sum by $1$, following Faulhaber's formulas.
Key Insight
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.