數學知識交流---分數和(2)

這兩題都是幾何級數的和，公比 r < 1。
應用公式：幾何級數和 = a/(1 - r)

(1) 求2/13 + 2/13^2 + 2/13^3 + ...

首項 a = 2/13 及 公比 = 1/13

2/13 + 2/13^2 + 2/13^3 + ...
= (2/13) / [1 - (1/13)]
= (2/13) / (12/13)
= (2/13) * (13/12)
= 1/6


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(2) 求7/3 + 7/3^2 + 7/3^3 + ...

首項 a= 7/3 及 公比 =1/3

7/3 + 7/3^2 + 7/3^3 + ...
= (7/3) / [1 - (1/3)]
= (7/3) / (2/3)
= (7/3) * (3/2)
= 7/2

2011-06-17 18:14:14 補充：
另一做法：

(1)
S = 2/13 + 2/13^2 + 2/13^3 + ...
(1/13)S = 2/13^2 + 2/13^3 + ...

兩式相減：
[1 - (1/13)]S = 2/13
(12/13)S = 2/13
S = 1/6 ...... 答案

(2)
S = 7/3 + 7/3^2 + 7/3^3 + ...
(1/3)S = 7/3^2 + 7/3^3 + ...

兩式相減：
[1 - (1/3)]S = 7/3
(2/3)S = 7/3
S = 7/2 ...... 答案

LET A =2/13 + 2/13^2 + 2/13^3 + ...-------------1
13A=2+2/13 + 2/13^2 + 2/13^3 + ...-------------2
2-1,
13A-A=2
12A=2
A=1/6

THIS WAY ALSO OK