Algebric Sets

Algebra Level 4

$\{1,2\} , \{ 4,5,6 \} , \{9,10,11,12\} , \{16,17,18,19,20\} , \ldots$

The first $n$ positive integers are partitioned into sets as described above.

Let $S_{n}$ be the sum of the elements of the set at $n^{\text{th}}$ position. Find the value of $\sqrt{3} \dfrac{S_{50}}{\sqrt{S_{51}-S_{50}}}.$