Arithmetic Sequence

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Source: algebrica.org - CC BY-NC 4.0 https://algebrica.org/arithmetic-sequence/ Fetched from algebrica.org post 14634; source modified 2026-03-12T20:56:26.

What is an arithmetic sequence

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.

Let’s consider, for example, the sequence of non-negative even numbers:

An arithmetic sequence can also be defined using a recursive formula:

a_n = a_1 + n \cdot d \quad \text{where } a_1, d \in \mathbb{R}

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.