Bài tập 9 trang 135 SGK Toán 11 NC

Bài tập 9 trang 135 SGK Toán 11 NC

a)

\(\begin{array}{l}
0,444... = 0,4 + 0,04 + 0,004 + ...\\
 = \frac{4}{{10}} + \frac{4}{{{{10}^2}}} + \frac{4}{{{{10}^3}}} + ...\\
 = 4\left( {\frac{1}{{10}} + \frac{1}{{{{10}^2}}} + \frac{1}{{{{10}^3}}} + ...} \right)\\
 = 4.\frac{{\frac{1}{{10}}}}{{1 - \frac{1}{{10}}}} = \frac{4}{9}
\end{array}\)

b)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
0,2121 = 0,21 + 0,0021 + ...\\
 = \frac{{21}}{{{{10}^2}}} + \frac{{21}}{{{{10}^4}}} + ...
\end{array}\\
{ = 21.\left( {\frac{1}{{{{10}^2}}} + \frac{1}{{{{10}^4}}} + ...} \right)}\\
{ = 21.\frac{{\frac{1}{{{{10}^2}}}}}{{1 - \frac{1}{{{{10}^2}}}}} = \frac{{21}}{{90}} = \frac{7}{{33}}}
\end{array}\)

c)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
0,32111... = \frac{{32}}{{100}} + \frac{1}{{1000}}\\
 + \frac{1}{{1000}}.\frac{1}{{10}} + \frac{1}{{1000}}.{\left( {\frac{1}{{10}}} \right)^2} + ...
\end{array}\\
\begin{array}{l}
 = \frac{{32}}{{100}} + \frac{1}{{1000}}.\frac{1}{{1 - \frac{1}{{10}}}}\\
 = \frac{{32}}{{100}} + \frac{1}{{900}} = \frac{{289}}{{900}}
\end{array}
\end{array}\)

-- Mod Toán 11