1000 followers problem - (2) (Easier than the first)

Algebra Level 4

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

I define three kinds of very similar functions as above.

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


Try the first problem here.
Try more of my originals in this set.
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