---
tag: 0MNM54WB99
kind: definition
name: Geometric sequence
aliases: [geometric-progression]
msc: [40A05]
defines: 0D1NY1ARDR
uses: [0CTV783HYT]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_progression
    locator: lead section
---

## Definition

Let $a$ and $r$ be real numbers.
The geometric sequence with initial term $a$ and common ratio $r$ is the [[real sequence|0CTV783HYT]] $(a\,r^n)_{n \in \mathbb{N}}$.
Its terms are $a, a r, a r^2, a r^3, \dots$, and each term is $r$ times the term before it.

## Remarks

Many texts require $a \ne 0$ and $r \ne 0$, so that the quotient of two consecutive terms is always $r$.
The sequence $1, 2, 4, 8, \dots$ has $a = 1$ and $r = 2$. The sequence $1, 1/2, 1/4, \dots$ has $r = 1/2$.
For $|r| < 1$, the sequence [[converges to $0$|0Q7K4AXE53]]. For $r > 1$ and $a > 0$, it [[tends to infinity|0TZK91NQE3]].
The sum of the terms is the [[geometric series|01X5CVV1Q5]].

## Explanations

### L1 human

In a geometric sequence, you go from one term to the next by multiplication with the same number $r$.
If $r = 3$ and you start at $2$, you get $2, 6, 18, 54, \dots$.