Arithmetic Series¶

Definition¶

Let $(a_i)_{i=1}^n$ be a sequence of real numbers defined by $a_i=a+(i-1)d$ where $a,d\in\mathbb{R}\text{.}$ This is known as an arithmetic sequence/progression and $d$ is called the common difference. The (finite) arithmetic series formula is given by

\sum_{i=1}^n (a+(i-1)d) =\frac{n(2a+(n-1)d)}{2} =na+\frac{n(n-1)d}{2}.

Interpretation¶

An arithmetic series is the sum of equally spaced terms: each term is the previous one plus the common difference $d$ . Its closed form gives the sum without adding the terms one by one.

A 6 by 7 rectangle of squares split into two identical staircases, one of 1 + 2 + ... + 6 squares and the same staircase flipped, showing that twice the sum equals 6 times 7.

Two copies of the staircase $1+2+\cdots+6$ $6$ , one blue and one red and turned around, fill a rectangle of $6\times7=42$ squares, so $1+2+\cdots+6=21$ $21$ . The same picture works for every $n$ : twice the sum is $n(n+1)$ $1)$ .

The trick behind the figure gives the general formula. Writing the sum forwards and backwards and adding term by term pairs the first term with the last, the second with the second-to-last, and so on, and every pair has the same sum $a+\bigl(a+(n-1)d\bigr)=2a+(n-1)d$ $1) d$ . There are $n$ pairs, so the series equals $n$ times the average of its first and last terms. The formula can also be proved by mathematical induction.

Sums of this kind appear in Riemann sums: with $n$ equal pieces and right endpoints, the Riemann sum of $f(x)=x$ on $[0,1]$ is $\displaystyle \sum_{i=1}^n\frac{i}{n}\cdot\frac{1}{n}=\frac{n+1}{2n}$ $\frac{n + 1}{2 n}$ , which tends to $\tfrac{1}{2}$ . An infinite arithmetic series diverges unless $a=d=0$ $0$ , because its terms do not tend to $0$ .

A common slip is an off-by-one count. The $n$ -th term is $a+(n-1)d$ $1) d$ , not $a+nd$ $n d$ , and a sum from $i=m$ to $i=n$ has $n-m+1$ terms.

Key Results¶

The sum of a finite arithmetic sequence can be expressed using only the first term, the common difference, and the number of terms.
Arithmetic series grow quadratically in the number of terms when the common difference is nonzero.

Examples¶

$\displaystyle \sum_{i=1}^{10} i = 55$
$\displaystyle \sum_{i=1}^{5} (3 + 2(i-1)) = 35$

Prerequisites¶

Finite Series

Sources¶

Math I: Calculus · Lecture 10

Keywords: arithmetic progression Symbols: element of, real numbers, sum

Last updated: 22 October 2026 19:04