Where's my Telescope?

Algebra Level 4

\[\begin{align} S & = \frac 1{1\times 2} +\frac 1{3\times 4} +\frac 1{5\times 6} + \cdots + \frac 1{99\times 100} \\ T & = \frac 1{51\times 100} + \frac 1{52\times 99 } + \frac 1{53\times 98} + \cdots + \frac 1{100\times 51} \end{align} \]

If \(\dfrac ST = \dfrac pq \), where \(p\) and \(q\) are coprime positive integers. Find \(p+q\).

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