Arithmetic Sequence

analysis

Definition

Arithmetic Sequence

A sequence $(a_n{)}_{n\ge 0}$ is called arithmetic if the difference between any two consecutive terms remains constant.

Recurrence relation:

$$a_{n+1}=a_n+d$$

Explicit formula:

$$a_n=a_0+nd$$

where $a_0$ is the first term and $d$ is the common difference.

Sum of an Arithmetic Sequence

The sum of the first $n+1$ terms of an arithmetic sequence is called an arithmetic series:

$$S_n=\sum _{k=0}^n(a_0+kd)=\frac{(n+1)(a_0+a_n)}{2}$$

Properties

• Constant Difference: $a_{n+1}-a_n=d$ for all $n$.
• Arithmetic Mean: Each term (except the first) is the arithmetic mean of its neighbours: $a_n=\frac{a_{n-1}+a_{n+1}}{2}$.
• Relation to Linear Functions: An arithmetic sequence can be viewed as a linear function $f(n)=dn+a_0$ sampled at integer values.