數學嘅數
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基本
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延伸
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其他
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圓周率 π = 3.141592653…
自然對數嘅底 e = 2.718281828…
虛數單位 i = ![{\displaystyle +{\sqrt {-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/126c0c912277f7c0b633ca5647270eb3d430342c)
無窮大量 ∞
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等比數列,有時叫幾何數列,係一種常見數列,每一項同前項嘅比係同一樣嘅,叫做公比。例如 1,2,4,8,16,32,...,公比係2。
- 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比都係同一個數,叫做公比,通常用
表示。
- 如果數列係
,
- 公比
![{\displaystyle r={a_{1} \over a_{0}}={a_{2} \over a_{1}}={a_{3} \over a_{2}}=...={a_{n} \over a_{n-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/437b5ec6526969898c18e262008b8300a1d5f6c2)
- 如果
係首項,就有:
![{\displaystyle a_{1}=a_{0}\ r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69ed0e082dbf9d9161420b9e5b288eb710977245)
![{\displaystyle a_{2}=a_{1}\ r=a_{0}\ r^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34c38ecdddfb20fc777851ea2045150fb9f09d30)
![{\displaystyle a_{3}=a_{2}\ r=a_{1}\ r^{2}=a_{0}\ r^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed23df56133bbd235ead07e8bb27d80e5dcb1f6)
![{\displaystyle ...}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00f2f395950e3698a46501d1e9aae8e8defa145)
![{\displaystyle a_{n}=a_{0}\ r^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df6084128a21530a13afaec473d88f4a32963ace)
假設
係頭
項嘅和:
,
根據定義可寫成
![{\displaystyle s_{n}=a_{0}+a_{0}\ r+a_{0}\ r^{2}+a_{0}\ r^{3}+...+a_{0}\ r^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56e2b2daf29874ad8fee947becb531a65215d182)
成條式除以
(按照定義
所以可當做除數 )
![{\displaystyle {s_{n} \over a_{0}}=1+r+r^{2}+r^{3}+...+r^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c72cf7be351cd1f713d3163f2533b5ae38e0111)
將等號右邊因式分解
![{\displaystyle {s_{n} \over a_{0}}={1-r^{n} \over 1-r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0600aa63acb45f3cf9fb1cc2dfbf7bc4685b35)
所以頭
項嘅和係
![{\displaystyle s_{n}=a_{0}{1-r^{n} \over 1-r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7aaf2c47569708f3492851f264c4d6f781778f45)
如果有無限項
![{\displaystyle {\lim _{n\to \infty }}s_{n}={\lim _{n\to \infty }}\left(a_{0}{1-r^{n} \over 1-r}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa7790d9f8905c87e37a34ee8f5cc6d946522fe)
當
時收斂
![{\displaystyle {\lim _{n\to \infty }}s_{n}={a_{0} \over 1-r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df496647fc0ec86b1c8b713b8ea272436f4bf63b)