# 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 $$U_n>201600$$.

×