為什麼這篇無窮等比級數和鄉民發文收入到精華區：因為在無窮等比級數和這個討論話題中，有許多相關的文章在討論，這篇最有參考價值！作者yueayase (scrya)看板Math標題Re: [中學] 無窮等比級數時間Wed Apr...
※ 引述《ChenYM (老宅男一個)》之銘言：
: https://i.imgur.com/0MJA3K7.jpg
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令a 公比r b 公比r
n 1 n 2
兩個首項都為1 => a = b = 1
1 1
=>Σa = 1/(1-r ), Σb = 1/(1-r )
n 1 n 2
因為a 和b 都為等比數列，所以a b 也是等比數列，且首項為1*1=，1公比為r r
n n n n 1 2
=>Σa b = 1/(1-r r )
n n 1 2
由題目設定，
1/(1-r r ) = 3/4 => 1-r r = 4/3 => r r = -1/3
1 2 1 2 1 2
Σ(a + b ) = Σa + Σb = 1/(1-r ) + 1/(1-r ) = 13/5
n n n n 1 2
(1-r ) +(1-r ) (2-r -r ) 2 - r - r
2 1 1 2 1 2
=> -------------- = ------------ = ------------- = 13/5
(1-r )(1-r ) 1-r -r +r r 2/3 -r -r
1 2 1 2 1 2 1 2
=> 5(2-r - r ) = 13(2/3-r - r )
1 2 1 2
=> 15[2-(r +r )] = 13[2-3(r + r )]
1 2 1 2
=> 24(r + r ) = -4
1 2
=> r + r = -1/6
1 2
2 2
=> (r -r ) = (r +r ) -4r r = 1/36 + 4/3 = 49/36
1 2 1 2 1 2
=> r -r = ±7/6
1 2
若r - r = 7/6 => 2r = 1 => r = 1/2, r = -1/6 - 1/2 = -4/6 = -2/3
1 2 1 1 2
若r - r = -7/6 => 2r = -8/6 => r = -2/3, r = -1/6 + 2/3 = 3/6 = 1/2
1 2 1 1 2
=> 兩公比為1/2和-2/3
2 2
因為a 和 b 為等比數列，所以a 和b 也為等比數列
n n n n
2 2 2
且首項皆為1 = 1，公比各自為r 和r
1 2
2 2 2 2 2
∴Σ(a + b ) = Σa +2Σa b +Σb = 1/(1-r ) + 2*3/4 + 1/(1-r )
n n n n n n 1 2
= 1/(1-1/4) + 3/2 + 1/(1-4/9)
= 4/3 + 3/2 + 9/5
= 40/30 + 45/30 + 54/30
= 139/30
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