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107全國高中聯招

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填充2.
有一數列\(\langle\;a_n\rangle\;\),若\(a_1=0\),且\(a_{n+1}-1=a_n+2\sqrt{1+a_n}\),\(n=1,2,3,\ldots\),求\(a_{30}=\)   
[解答]
\(\begin{align}
  & {{a}_{n+1}}-1=a{}_{n}+2\sqrt{1+{{a}_{n}}} \\
& {{a}_{n+1}}+1=a{}_{n}+1+2\sqrt{1+{{a}_{n}}}+1={{\left( \sqrt{a{}_{n}+1}+1 \right)}^{2}} \\
& {{\left( \sqrt{a{}_{n+1}+1} \right)}^{2}}={{\left( \sqrt{a{}_{n}+1}+1 \right)}^{2}} \\
& \sqrt{a{}_{n+1}+1}=\sqrt{a{}_{n}+1}+1 \\
& \cdots \cdots  \\
\end{align}\)

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