2008-4 Limit (9/25)

Let \(a_1=\sqrt{1+2}\),
\(a_2=\sqrt{1+2\sqrt{1+3}}\),
\(a_3=\sqrt{1+2\sqrt{1+3\sqrt{1+4}}}\), …,
\(a_n=\sqrt{1+2\sqrt{1+3\sqrt {\cdots \sqrt{\sqrt{\sqrt{\cdots\sqrt{1+n\sqrt{1+(n+1)}}}}}}}}\), … .

Prove that \(\displaystyle\lim_{n\to \infty} \frac{a_{n+1}-a_{n}}{a_n-a_{n-1}}=\frac12\).

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