Dual summation

Calculus Level 4

\[\sum_{i=1}^{1728}(i)(i-1727)=N\] and \[\sum_{e=1}^{\infty}\frac{6e+7}{7^{e+1}(e)(e+1)}=M\] . Then find the remainder when \(\frac{MN+1}{M}\) is divided by \(1000\).

Problem is original.

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