$\begin{aligned} S_{n}&=&\displaystyle\sum_{i=1}^{n} a_i \\ S_{n}^{'}&=&\displaystyle \sum_{i=1}^{n} \dfrac{1}{a_i} \\ S(n)&=&\displaystyle\sum_{i=1}^{n} (S_{n}^{'}a_i - 1) \end{aligned}$

I define three kinds of very similar functions as above.

Find the value of $S(1000)-S_{1000}S_{1000}^{'}$ .

The answer is -1000.

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$S(n)=\displaystyle\sum_{i=1}^{n} (S_{n}^{'}a_i - 1) \\ \Rightarrow S(n)=\displaystyle\sum_{i=1}^{n} S_{n}^{'}a_{i} - \displaystyle\sum_{i=1}^{n} 1 \\ \Rightarrow S(n)=\displaystyle\sum_{i=1}^{n} S_{n}^{'}a_{i} - n \\ \Rightarrow S(n)= S_{n}^{'}a_{1} + S_{n}^{'}a_{2}+ S_{n}^{'}a_{3} \dots S_{n}^{'}a_{n} - n \\ \Rightarrow S(n)= S_nS_{n}^{'}-n \\ \Rightarrow S(n)-S_nS_{n}^{'}=-n \\ \Rightarrow S(1000)-S_{1000}S_{1000}^{'} = \boxed{-1000}$