Harmonically Complex!

$\begin{aligned} S & = \sum_{i=1}^{200} (-1)^i \frac {H_i + H_{i+1}}{H_i - H_{i+1}} \quad \quad \small \color{#3D99F6} \text{Let } H_i = \frac 1{a+(i-1)d} \text{, where } a = \text{ first term and } d= \text{ common difference.} \\ & = \sum_{i=1}^{200} (-1)^i \frac {\frac 1{a+(i-1)d} + \frac 1{a+id}}{\frac 1{a+(i-1)d} - \frac 1{a+id}} \\ & = \sum_{i=1}^{200} (-1)^i \frac {2a+2id-d}d \\ & = \sum_{i=1}^{200} (-1)^i \left(\frac {2a}d + 2i -1 \right) \\ & = \left(\frac {2a}d -1 \right) {\color{#3D99F6} \sum_{i=1}^{200} (-1)^i} + 2 \sum_{i=1}^{200} (-1)^i i \\ & = \left(\frac {2a}d -1 \right) {\color{#3D99F6} (0)} + 2 \sum_{i=1}^{200} (-1)^i i \\ & = 2 ({\color{#3D99F6}-1+2}{\color{#D61F06}-3+4}{\color{#3D99F6}-5+6}-\cdots{\color{#D61F06}-199+200}) \\ & = 2 \underbrace{({\color{#3D99F6}1} + {\color{#D61F06}1}+{\color{#3D99F6}1}+\cdots+{\color{#D61F06}1})}_{100 \times 1} \\ & = \boxed{200} \end{aligned}$