Arithmetic and geometric sequences

Among the sequences defined by recurrence, two families are so frequent that they have a name of their own: those in which at each step a constant is added and those in which at each step there is a multiplication by a constant.

Example — Arithmetic and geometric sequence

• Arithmetic with first term $a_0$ and common difference $d$:   $a_{n+1} = a_n + d$,   from which $a_n = a_0 + n\,d$.
• Geometric with first term $a_0$ and common ratio $q$:   $a_{n+1} = q\,a_n$,   from which $a_n = a_0\,q^n$.

In the arithmetic sequence the term grows (or decreases) by a fixed quantity at each step; in the geometric one it is repeatedly multiplied by a fixed factor. These two families have historical names and well-known closed formulae, which are formalised in the following section.

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Topics: Sequences
Concepts: Arithmetic progression · Geometric progression · Common ratio · Sequence by recurrence