# 等比數列

${\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} }$

## 定義

• 數列嘅每一項都唔可以係0。
• 除咗首項，數列嘅每一項同前項嘅都係同一個數，叫做公比，通常用 ${\displaystyle r}$ 表示。
• 如果數列係 ${\displaystyle a_{0}\ ,\ a_{1}\ ,\ a_{2}\ ,\ ...\ ,\ a_{n}}$

• 如果 ${\displaystyle a_{0}}$ 係首項，就有：
${\displaystyle a_{1}=a_{0}\ r}$
${\displaystyle a_{2}=a_{1}\ r=a_{0}\ r^{2}}$
${\displaystyle a_{3}=a_{2}\ r=a_{1}\ r^{2}=a_{0}\ r^{3}}$
${\displaystyle ...}$
${\displaystyle a_{n}=a_{0}\ r^{n}}$

## 求和公式

${\displaystyle s_{n}={\sum _{k=0}^{n-1}}\ a_{k}=a_{0}+a_{1}+a_{2}+...+a_{n-1}}$

${\displaystyle s_{n}=a_{0}+a_{0}\ r+a_{0}\ r^{2}+a_{0}\ r^{3}+...+a_{0}\ r^{n-1}}$

${\displaystyle {s_{n} \over a_{0}}=1+r+r^{2}+r^{3}+...+r^{n-1}}$

${\displaystyle {s_{n} \over a_{0}}={1-r^{n} \over 1-r}}$

${\displaystyle s_{n}=a_{0}{1-r^{n} \over 1-r}}$

${\displaystyle {\lim _{n\to \infty }}s_{n}={\lim _{n\to \infty }}\left(a_{0}{1-r^{n} \over 1-r}\right)}$

${\displaystyle |r|<1}$ 時收斂

${\displaystyle {\lim _{n\to \infty }}s_{n}={a_{0} \over 1-r}}$