Definition

Definition 1. A sequence $a_n$ is called an arithmetic sequence (or arithmetic progression) if it consists of numbers arranged in such a way that the difference between any term and the one before it is constant. It is characterized by terms of the form:

$$a_1, a_2, \dots, a_n \quad \text{with} \quad a_n - a_{n-1} = d$$

  • By convention, the first term of an arithmetic progression is typically indexed with $n = 1.$
  • $d$ represents the difference between two consecutive terms in an arithmetic progression, and it is known as the common difference.
  • If $d > 0$, the progression is increasing.
  • If $d < 0$, the progression is decreasing.
  • If $d = 0$, the progression is constant.

Consider, for example, the sequence of non-negative even numbers:

$$0, 2, 4, 6, 8, 10, \dots$$

The first term is $a_1 = 0$ and the common difference is $d = 2$.

The non-negative even numbers form an arithmetic progression with a common difference d of 2. 2 4 6 8 0

Every term is obtained by adding $2$ to the preceding one, so the sequence satisfies the definition of an arithmetic progression.

An arithmetic sequence can also be defined recursively, by prescribing the first term and the rule that produces each subsequent term from the previous one:

$$ \begin{cases} a_1 = a \\[0.5em] a_n = a_{n-1} + d \quad \text{for } n \geq 2 \end{cases} $$

with $a, d \in \mathbb{R}$.

2 6 10 14 1 3 5 n (index) aₙ An arithmetic sequence grows through constant increments between consecutive terms.

An arithmetic progression exhibits a characteristic stepwise pattern, where the height of each step corresponds to the common difference between consecutive terms in the sequence.

In an arithmetic progression, each term $a_n$ is obtained by adding the first term $a_1$ to the product of the common difference $d$ and $(n - 1)$. This gives the general formula for the $n$-th term:

$$ a_n = a_1 + (n - 1) \cdot d \quad \text{for } n \geq 1 $$

This formula allows you to compute any term in the sequence directly, without listing all the previous ones.

Example

Let's define an arithmetic sequence with first term $a_1 = 2$ and common difference $d = 3$. We use the formula:

$$ a_n = a_1 + (n - 1) \cdot d $$

Plug in the values:

$$ a_n = 2 + (n - 1) \cdot 3 $$

Now calculate the first few terms:

  • $a_1 = 2$
  • $a_2 = 2 + 1 \cdot 3 = 5$
  • $a_3 = 2 + 2 \cdot 3 = 8$
  • $a_4 = 2 + 3 \cdot 3 = 11$
  • $a_5 = 2 + 4 \cdot 3 = 14$

The resulting sequence is:

$$ 2,\ 5,\ 8,\ 11,\ 14,\ \dots $$

Sum of $n$ terms of an arithmetic progression

The sum $S_n$ of the first $n$ terms $a_1, a_2, \dots, a_n$ of an arithmetic progression is equal to the product of $n$ and the average of the first and last term:

$$ S_n = n \cdot \frac{a_1 + a_n}{2} $$

This formula allows you to quickly compute the total sum of a finite number of terms in an arithmetic progression. For example, consider the arithmetic progression of non-negative even numbers:

$$ 2,\ 4,\ 6,\ 8,\ 10 $$

We want to calculate the sum of the first 5 terms $(n = 5).$ Using the formula, we have:

$$ S_5 = 5 \cdot \frac{2 + 10}{2} = 5 \cdot 6 = 30 $$

This illustrates the same reasoning behind Gauss's trick: by pairing the first and last terms, you can quickly compute the total sum of an arithmetic progression. The closed form of $S_n$ can also be established via the principle of mathematical induction, with $n = 1$ as the base case.

Derivation of the sum formula

The closed-form expression for $S_n$ can be obtained by a symmetry argument that does not require induction. Write the sum twice, the second time in reverse order, and align the two rows term by term:

$$ \begin{aligned} S_n &= a_1 + a_2 + a_3 + \cdots + a_{n-1} + a_n \\[6pt] S_n &= a_n + a_{n-1} + a_{n-2} + \cdots + a_2 + a_1 \end{aligned} $$

Each column contains a pair of terms that, by definition of arithmetic progression, has constant sum $a_1 + a_n$. Indeed, if $a_k = a_1 + (k-1) d$, then:

$$ a_k + a_{n-k+1} = \bigl(a_1 + (k-1)d\bigr) + \bigl(a_1 + (n-k)d\bigr) = 2a_1 + (n-1)d = a_1 + a_n $$

Summing the two rows column by column gives $2 S_n = n (a_1 + a_n)$, from which the formula:

$$ S_n = n \cdot \frac{a_1 + a_n}{2} $$

follows directly. Substituting $a_n = a_1 + (n-1)d$ yields the alternative expression:

$$ S_n = \frac{n}{2}\bigl(2 a_1 + (n-1) d\bigr) $$

which depends only on the first term, the common difference, and the number of terms.

Connection with linear functions

The general term $a_n = a_1 + (n-1) d$ is a linear function of the index $n$, with slope equal to the common difference $d$ and intercept $a_1 - d$. The graph of the sequence consists of evenly spaced points lying on a straight line, in contrast to the geometric sequence, whose terms lie on the graph of an exponential function. This linear nature explains why an arithmetic progression models every quantity that grows or decreases by a fixed amount per unit step, from simple interest at a constant rate to the partial sums of a constant series.