An algebra problem by Rocco Dalto

$For$ $any$ $positive$ $integer$ $J$ $using$ $partial$ $fractions$ $we$ $have:$ $$ $\frac{1}{(X - J) * (X - J -1)} =$ $\frac{A}{X - J} + \frac{B}{X - J - 1}$ $\implies$
$A + B = 0$ $(J + 1) * A + J * B = -1$ $\implies$ $A = -1$ $and$ $B = 1$ $\implies$ $\frac{1}{(X - J) * (X - J -1)} =$ $\frac{1}{X - J - 1} - \frac{1}{X - J}$ $$ $\therefore$ $for$ $any$ $positive$ $integer$ $N$ $we$ $have:$ $$ $\sum\limits_{j = 1}^N \frac{1}{(X - J) * (X - J -1)}$ $=$ $\frac{1}{X - N - 1} - \frac{1}{X - 1}$ = $\frac{1}{M},$ $$ $$ $where$ $M > 0.$

$For$ $M > 0$ $we$ $have$ $the$ $real$ $solutions:$

$X_1,X_2 = \frac{(N + 2) \pm \sqrt{N * (N + 4 * M)}}{2}$ $and$ $X_1 + X_2 = N + 2$ $$ $$ $$ $\therefore$ $$ $N = 1000$ $\implies$ $X_1 + X_2 = 1002.$