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

The best solution was submitted by Jaehoon Kim (김재훈), 수리과학과 2003학번. Congratulations! 

Here is his Solution of Problem 2008-4.