Arithmetic Sequence

What is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where the difference between each pair of consecutive numbers is always the same. For example, in $2, 4, 6, 8$ , the difference is $2$ .

General Form of an Arithmetic Sequence

An arithmetic sequence can be written as: $a, \, a + d, \, a + 2d, \, a + 3d, \, \dots$

where:

$a$ is the first term,
$d$ is the common difference, and
each term is calculated by adding $d$ to the previous term.

For example, in the sequence $5, 8, 11, 14, \dots$ :

The first term is $a = 5$ ,
The common difference is $d = 3$ , because $8 - 5 = 3$ and $11 - 8 = 3$ .

Key Characteristics

Common Difference: The common difference $d$ is found by subtracting any term from the term that follows it: $d = a_n - a_{n-1}$ where $a_n$ is the $n$ -th term and $a_{n-1}$ is the term before it.
Formula for the $n$ -th Term: The -th term of an arithmetic sequence is given by:
- where:
  - $a_n$ is the $n$ -th term,
  - $a$ is the first term,
  - $d$ is the common difference, and
  - $n$ is the position of the term in the sequence.
- Example: For the sequence :
  - $a = 3$ , $d = 4$ ,
  - To find the $10$ -th term:
  - $a_{10} = 3 + (10 - 1)(4) = 3 + 36 = 39.$

Applications of Arithmetic Sequences

Real-Life Examples:
- Counting patterns, such as even numbers ( $2, 4, 6, 8, \dots$ ) or odd numbers ( $1, 3, 5, 7, \dots$ ).
- Incremental savings: If you save a fixed amount each week, the total savings follows an arithmetic sequence.
Mathematical Applications:
- Used in summation problems and number patterns.
- Helpful in solving problems involving linear relationships in algebra and geometry.