Tìm n: \(\frac{1}{{{\log }_{2}}x}+\frac{1}{{{\log }_{{{2}^{2}}}}x}+\frac{1}{{{\log }_{{{2}^{3}}}}x}+..

* Hướng dẫn giải

\(\frac{1}{{{{\log }_2}x}} + \frac{1}{{{{\log }_{{2^2}}}x}} + \frac{1}{{{{\log }_{{2^3}}}x}} + ... + \frac{1}{{{{\log }_{{2^n}}}x}} = {\log _x}2 + {\log _x}{2^2} + {\log _x}{2^3} + ... + {\log _x}{2^n}\)

\( = {\log _x}\left( {{{2.2}^2}{{.2}^3}{{...2}^n}} \right) = 465{\log _x}2 = {\log _x}{2^{465}}\)

\( \Rightarrow {2.2^2}{.2^3}{...2^n} \Leftrightarrow 1 + 2 + 3 + ... + n = 465 \Leftrightarrow \frac{n}{2}\left( {n + 1} \right) = 465\)

\( \Leftrightarrow {n^2} + n - 930 = 0 \Leftrightarrow \left[ \begin{array}{l} n = 30\\ n = - 31 \end{array} \right. \Rightarrow n = 30\)