Cho dãy số (un) là một cấp số nhân có số hạng đầu u1 = 1, công bội q = 2 .

* Hướng dẫn giải

\(\begin{array}{l}
T = \frac{1}{{{u_1} - {u_5}}} + \frac{1}{{{u_2} - {u_6}}} + \frac{1}{{{u_3} - {u_7}}} + ... + \frac{1}{{{u_{20}} - {u_{24}}}}\\
 = \frac{1}{{{u_1}\left( {1 - {q^4}} \right)}} + \frac{1}{{{u_2}\left( {1 - {q^4}} \right)}} + \frac{1}{{{u_3}\left( {1 - {q^4}} \right)}} + ... + \frac{1}{{{u_{20}}\left( {1 - {q^4}} \right)}}\\
 = \frac{1}{{1 - {q^4}}}\left( {\frac{1}{{{u_1}}} + \frac{1}{{{u_2}}} + \frac{1}{{{u_3}}} + ... + \frac{1}{{{u_{20}}}}} \right)\\
 = \frac{1}{{1 - {q^4}}}\left( {\frac{1}{{{u_1}}} + \frac{1}{{{u_1}q}} + \frac{1}{{{u_1}{q^2}}} + ... + \frac{1}{{{u_1}{q^{19}}}}} \right)\\
 = \frac{1}{{1 - {q^4}}}.\frac{1}{{{u_1}}}\left( {1 + \frac{1}{q} + \frac{1}{{{q^2}}} + ... + \frac{1}{{{q^{19}}}}} \right)\\
 = \frac{1}{{1 - {q^4}}}.\frac{1}{{{u_1}}}.\frac{{{{\left( {\frac{1}{q}} \right)}^{20}} - 1}}{{\frac{1}{q} - 1}} = \frac{1}{{1 - {q^4}}}.\frac{1}{{{u_1}}}.\frac{{1 - {{\left( q \right)}^{20}}}}{{\left( {1 - q} \right){q^{19}}}} = \frac{{1 - {2^{20}}}}{{{{15.2}^{19}}}}
\end{array}\)