# Three Sums

**Algebra**Level 5

For a positive integer \(k\), let \(x_k\) be the largest positive real solution to the equation
\[\frac{2}{x}+\frac{1}{\sqrt{k(k+1)x^2-1}}=1.\]
Let
\[S_n=\sum_{k=1}^n\frac{1}{\sqrt{2+2x_k}}, \quad T_n=\sum_{k=1}^n\frac{1}{1+\sqrt{2}+2S_k}\]

and
\[U_n=\sum_{k=1}^n\left(T_k^2+\frac{k^2}{T_k^2}\right).\]
Find the least positive integer \(n\) such that \(U_n>201600\).