Definition
Arithmetic Sequence
A sequence
is called arithmetic if the difference between any two consecutive terms remains constant. Recurrence relation:
Explicit formula:
where
is the first term and is the common difference.
Sum of an Arithmetic Sequence
The sum of the first
Properties
- Constant Difference:
for all . - Arithmetic Mean: Each term (except the first) is the arithmetic mean of its neighbours:
. - Relation to Linear Functions: An arithmetic sequence can be viewed as a linear function
sampled at integer values.