Tính S = 2+3+ 4+ ..

\(S + 99 = \left( {\frac{{2 + 3 + 4 + ... + 100}}{1} + 1} \right) + \left( {\frac{{3 + 4 + ... + 100}}{{1 + 2}} + 1} \right) + .. + \left( {\frac{{100}}{{1 + 2 + ... + 99}} + 1} \right)\)

\(S + 99 = \frac{{1 + 2 + 3 + ... + 100}}{1} + \frac{{1 + 2 + 3 + ... + 100}}{{1 + 2}} + .. + \frac{{1 + 2 + 3 + ... + 100}}{{1 + 2 + 3 + ... + 99}}\)

\( = \left( {1 + 2 + ... + 100} \right) \times \left( {\frac{1}{1} + \frac{1}{3} + \frac{1}{6} + ... + \frac{1}{{99 \times 100:2}}} \right)\)

\(\left( {S + 99} \right) \times \frac{1}{2} = \left( {100 \times 101:2} \right) \times \left( {\frac{1}{{1 \times 2}} + \frac{1}{{2 \times 3}} + \frac{1}{{3 \times 4}} + ... + \frac{1}{{99 \times 100}}} \right)\)

\( = 5050 \times \left( {1 - \frac{1}{{100}}} \right)\)

\( = 5050 \times \frac{{99}}{{100}} = \frac{{9999}}{2}\)

\(S = \frac{{9999}}{2} \times 2 - 99 = 9900\)