A probability problem by atishay jain

The maximum number of pieces of pizza from $n$ cuts is given by (see Cirle Division by Lines ) $f(n) = \dfrac 12 (n^2+n+2)$ . Then we have:

$\begin{aligned} S & = \sum_{n=1}^{100} \frac 12(n^2+n+2) \\ & = \frac 12 \sum_{n=1}^{100} (n^2+n+2) \\ & = \frac 12 \left(\frac {100\times101\times 201}6 + \frac {100\times 101}2 + 2(100) \right) \\ & = \boxed{171800} \end{aligned}$