Tính \[S = \frac{1}{{1.4}} + \frac{1}{{4.7}} + \frac{1}{{7.10}} + ... + \frac{1}{{94.97}} + \frac{1}{{97.100}}\]

Hướng dẫn giải:

\[S = \frac{1}{{1.4}} + \frac{1}{{4.7}} + \frac{1}{{7.10}} + ... + \frac{1}{{94.97}} + \frac{1}{{97.100}}\]

\[3S = \frac{3}{{1.4}} + \frac{3}{{4.7}} + \frac{3}{{7.10}} + ... + \frac{3}{{94.97}} + \frac{3}{{97.100}}\]

\[3S = \frac{{4 - 1}}{{1.4}} + \frac{{7 - 4}}{{4.7}} + \frac{{10 - 7}}{{7.10}} + ... + \frac{{97 - 94}}{{94.97}} + \frac{{100 - 97}}{{97.100}}\]

\[3S = \left( {\frac{4}{{1.4}} - \frac{1}{{1.4}}} \right) + \left( {\frac{7}{{4.7}} - \frac{4}{{4.7}}} \right) + \left( {\frac{{10}}{{7.10}} - \frac{7}{{7.10}}} \right) + ... + \left( {\frac{{97}}{{94.97}} - \frac{{94}}{{94.97}}} \right) + \left( {\frac{{100}}{{97.100}} - \frac{{97}}{{97.100}}} \right)\]

\(3S = \left( {1 - \frac{1}{4}} \right) + \left( {\frac{1}{4} - \frac{1}{7}} \right) + \left( {\frac{1}{7} - \frac{1}{{10}}} \right) + ... + \left( {\frac{1}{{94}} - \frac{1}{{97}}} \right) + \left( {\frac{1}{{97}} - \frac{1}{{100}}} \right)\)

\(3S = 1 - \frac{1}{4} + \frac{1}{4} - \frac{1}{7} + \frac{1}{7} - \frac{1}{{10}} + ... + \frac{1}{{94}} - \frac{1}{{97}} + \frac{1}{{97}} - \frac{1}{{100}}\)

\(3S = 1 - \frac{1}{{100}}\)

\(3S = \frac{{100}}{{100}} - \frac{1}{{100}}\)

\(3S = \frac{{99}}{{100}}\)

Suy ra \(S = \frac{{99}}{{100}}:3\)

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

\(S = \frac{{33}}{{100}}\)

Vậy \(S = \frac{{33}}{{100}}\)