Tìm x, biết : \(\left( {{{11} \over {12}} + {{11} \over {12.23}} + {{11} \over {23.34}} + ... + {{11} \over {89.100}}} \right) + x = {5 \over 3}\)

* Hướng dẫn giải

\(\eqalign{  & \left( {{{11} \over {12}} + {{11} \over {12.23}} + {{11} \over {23.34}} + ... + {{11} \over {89.100}}} \right) + x = {5 \over 3}  \cr  & {{11} \over {12}} + {{23 - 12} \over {12.23}} + {{34 - 23} \over {34.23}} + ... + {{100 - 89} \over {89.100}} + x = {5 \over 3}  \cr  & {{11} \over {12}} + {1 \over {12}} - {1 \over {23}} + {1 \over {23}} - {1 \over {34}} + ... + {1 \over {89}} - {1 \over {100}} + x = {5 \over 3}  \cr  & {{11} \over {12}} + {1 \over {12}} - {1 \over {100}} + x = {5 \over 3} \Leftrightarrow 1 - {1 \over {100}} + x = {5 \over 3}  \cr  & {{99} \over {100}} + x = {5 \over 3} \Leftrightarrow x = {5 \over 3} - {{99} \over {100}} \Leftrightarrow x = {{500} \over {300}} - {{297} \over {300}} \Leftrightarrow x = {{203} \over {300}}. \cr} \)