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7.3. 理解等比數列的概念及其性質(Understand the ConCept and the Properties of Geometric Sequence)

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香港中學文憑 – 數學科 必修部份 非基礎課題 v1.2

中學文憑溫習室 http://www.takwing.idv.hk/dse_room

7.3. 理解等比數列的概念及其性質(Understand the ConCept and the Properties of Geometric Sequence)

7.3.1. 等比數列的概念

² 其實等比數列同等差數列嘅概念大同小異。

² 當中最唔同嘅就係:

n 等差數列入面項與項之間嘅差係一樣嘅(即有個“公差”存在)。

n 而喺等比數列入面,項與項之間嘅比就一樣。

u 依個“相等嘅比”叫做“公比”(Common Ratio),記作“R”或“r”。

u 通常我哋計公差嘅方法係將兩個連續項數相減(但留意係後面個數除前面個 數):

R = Tn / Tn-1

l 例子: 3, 6, 12, 24, 48, …… 係一個等比數列。

n 原因係每一項同前一項嘅比都係2。

7.3.2. 等比數列的性質 l T = T × T

n 依點係講緊喺等比數列入面:

u 對三個相連項數嚟講,“中間嗰個數嘅二次方”等於“前後兩個數乘埋嘅積”。

u 另一種講法係:三個相連項中間嗰個數會係“前後兩個數乘埋再開方”。

l 如果“T1,T,T,…”係個等比數列,咁“kT1 , kT,kT,…”都會係一個等比數列。

n 要理解依點其實大家先要記住喺等差數列入面,項與項之間嘅比係一樣嘅。

n 如果“T1,T,T,…”之間嘅比係 R 咁多,咁

u “kT1, kT,kT,…”之間嘅比都係 R(因為兩個數互除時,大家嘅 k 會約咗)。

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