a+b

student_qwh

zeroieme

算下前6项
\(1+\sqrt{5},-\frac{1}{\sqrt{5}},\frac{1}{4} \left(5-\sqrt{5}\right),1+\sqrt{5},-\frac{1}{\sqrt{5}},\frac{1}{4} \left(5-\sqrt{5}\right)\)
后面步骤你应当会了。

student_qwh

谢谢

mathematica

1. (*https://bbs.emath.ac.cn/thread-9736-1-2.html*)
   
2. Clear["Global`*"];(*Clear all variables*)
   
3. a=Sqrt[5]+1
   
4. list={a}
   
5. Do[a=FullSimplify[1/(1-a)];list=Append[list,a],{k,59}]
   
6. aa=Total@list
   
7. bb=Times@@list
   
8. cc=FullSimplify[aa+bb]
   

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表是
\[\left\{\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}},\sqrt{5}+1,-\frac{1}{\sqrt{5}},\frac{1}{1+\frac{1}{\sqrt{5}}}\right\}\]

求和结果是
\[16 \sqrt{5}+\frac{20}{1+\frac{1}{\sqrt{5}}}+20\]
求积的结果是
\[1\]
两者的和是
\[11 \sqrt{5}+46\]

zeroieme

1. Sqrt[5]+1//NestList[(1-#)^-1&,#,59]&//(Plus@@#)+(Times@@#)&//Simplify

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