各位高手：
請問這題要如何解？
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中大數學系大學甄選入學試題[103](2)
2015-04-06 8:02 am
回答 (3)
2015-04-06 9:16 am
✔ 最佳答案
令 S = F₀ + 3/5 F₁ + 3²/5² F₂+ ... + 3ⁿ/5ⁿ Fn + ... 則 3/5 S = 3/5 F₀ + 3²/5² F₁+ 3³/5³ F₂+ ... + 3ⁿ⁺¹/5ⁿ⁺¹ Fn + ... S + 3/5 S
= F₀ + 3/5 ( F₀ + F₁) + 3²/5² (F₁+ F₂) + 3³/5³ (F₂+ F₃) + ... + 3ⁿ/5ⁿ (Fn-1 + Fn) + ...8/5 S = F₀ + 3/5 F₂+ 3²/5² F₃+ 3³/5³ F₄+ ... + 3ⁿ/5ⁿ Fn+1 + ...8/5 S = F₁+ 3/5 F₂+ 3²/5² F₃+ 3³/5³ F₄+ ... + 3ⁿ/5ⁿ Fn+1 + ...(8/5 S)(3/5) = 3/5 F₁ + 3²/5² F₂+ 3³/5³ F₃+ ... + 3ⁿ/5ⁿ Fn + ... (8/5 S)(3/5) + F₀ = F₀ + 3/5 F₁ + 3²/5² F₂+ 3³/5³ F₃+ ... + 3ⁿ/5ⁿ Fn + ... 24/25 S + 1 = SS = 25
2015-04-06 5:51 pm
謝謝兩位知識長，我懂了！
2015-04-06 9:44 am
Sol
S=F(0)+3F(1)/5+3^2F(2)/5^2+3^3F(3)/5^3+3^4F(4)/5^4+…
S=1+3/5+3^2F(2)/5^2+3^3F(3)/5^3+3^4F(4)/5^4+…
3^2F(2)/5^2+3^3F(3)/5^3+3^4F(4)/5^4+…=S－8/5
3F(2)/5+3^2F(3)/5^2+3^3F(4)/5^3+…=5S/3－8/3
So
S=F(0)+3F(1)/5+3^2F(2)/5^2+3^3F(3)/5^3+3^4F(4)/5^4+…
3S/5= 3F(0)/5+3^2F(1)/5^2+3^3F(2)/5^3+3^4F(3)/5^4+…
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2015-04-06 01:44:56 補充：
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8S/5=F(0)+3F(2)/5+3^2F(3)/5^2+3^3F(4)/5^3+3^4F(5)/5^3+…
8S/5=1+5S/3－8/3
24S=15+25S－40
S=25
S=F(0)+3F(1)/5+3^2F(2)/5^2+3^3F(3)/5^3+3^4F(4)/5^4+…
S=1+3/5+3^2F(2)/5^2+3^3F(3)/5^3+3^4F(4)/5^4+…
3^2F(2)/5^2+3^3F(3)/5^3+3^4F(4)/5^4+…=S－8/5
3F(2)/5+3^2F(3)/5^2+3^3F(4)/5^3+…=5S/3－8/3
So
S=F(0)+3F(1)/5+3^2F(2)/5^2+3^3F(3)/5^3+3^4F(4)/5^4+…
3S/5= 3F(0)/5+3^2F(1)/5^2+3^3F(2)/5^3+3^4F(3)/5^4+…
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2015-04-06 01:44:56 補充：
---------------------------------------------------------------------------
8S/5=F(0)+3F(2)/5+3^2F(3)/5^2+3^3F(4)/5^3+3^4F(5)/5^3+…
8S/5=1+5S/3－8/3
24S=15+25S－40
S=25
收錄日期: 2021-04-21 22:31:01
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20150406000010KK00009