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 \[\sqrt{3} \dfrac{S_{50}}{\sqrt{S_{51}-S_{50}}}.\]

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