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$\frac{1}{{102 \times 101 \times 100}} + \frac{1}{{101 \times 100 \times 99}} + ......... + \frac{1}{{3 \times 2 \times 1}} = ?$

On rewriting we have the series as,

$\frac{1}{{1 \times 2 \times 3}} + \frac{1}{{2 \times 3 \times 4}} + ................. + \frac{1}{{100 \times 101 \times 102}}$

$ = \sum\limits_{r = 1}^{100} {\frac{1}{{r(r + 1)(r + 2)}}} $

$ = \frac{1}{2}\sum\limits_{r = 1}^{100} {\left[ {\frac{1}{{r(r + 1)}} - \frac{1}{{(r + 1)(r + 2)}}} \right]} $

$ = \frac{1}{2}\left[ {\left( {\frac{1}{{1 \times 2}} - \frac{1}{{2 \times 3}}} \right) + \left( {\frac{1}{{2 \times 3}} - \frac{1}{{3 \times 4}}} \right) + .............+\left( {\frac{1}{{100 \times 101}} - \frac{1}{{101 \times 102}}} \right)} \right]$

$ = \frac{1}{2}\left( {\frac{1}{{1 \times 2}} - \frac{1}{{101 \times 102}}} \right)$

$ = \frac{1}{4}\left( {1 - \frac{1}{{101 \times 51}}} \right)$

$ = \frac{1}{4}\left( {\frac{{101 \times 51 - 1}}{{101 \times 51}}} \right)$

$ = \frac{1}{4} \times \frac{{5150}}{{101 \times 51}} = \frac{{2575}}{{2 \times 101 \times 51}} = \frac{{2575}}{{10302}}$

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