Ejercicio 296 – Solucionario Baldor

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DESCARGA

CAPITULO XXXVII

Progresiones Geométricas
Valor de una fracción decimal periódica

Ejercicio 296

Hallar por la suma al infinito, el valor de las fracciones decimales:

0.66666…..
$\begin{matrix} 0.6666… & = \frac{6}{10} + \frac{6}{100} + \frac{6}{1000} + …. \\ D o n d e : \\ a = \frac{6}{10}, r = \frac{\frac{6}{100}}{\frac{6}{10}} = \frac{1}{10} \\ S & = \frac{a}{1 - r} \\ S & = \frac{\frac{6}{10}}{1 - \frac{1}{10}} \\ S & = \frac{\frac{}{10}}{\frac{}{10}} \\ S & = \frac{2}{3} \\ 0.6666…. & = \frac{2}{3} \end{matrix}$
0.1212…..
$\begin{matrix} 0.1212… & = \frac{12}{100} + \frac{12}{10000} + \frac{12}{1000000} + …. \\ D o n d e : \\ a = \frac{12}{100}, r = \frac{\frac{12}{10000}}{\frac{12}{100}} = \frac{1}{100} \\ S & = \frac{a}{1 - r} \\ S & = \frac{\frac{12}{100}}{1 - \frac{1}{100}} \\ S & = \frac{\frac{}{100}}{\frac{}{100}} \\ S & = \frac{4}{33} \\ 0.121212…. & = \frac{4}{33} \end{matrix}$
0.159159…..
$\begin{matrix} 0.159159… & = \frac{159}{1000} + \frac{159}{1000000} + …. \\ D o n d e : \\ a = \frac{159}{1000}, r = \frac{\frac{159}{1000000}}{\frac{159}{1000}} = \frac{1}{1000} \\ S & = \frac{a}{1 - r} \\ S & = \frac{\frac{159}{1000}}{1 - \frac{1}{1000}} \\ S & = \frac{\frac{}{1000}}{\frac{}{1000}} \\ S & = \frac{53}{333} \\ 0.159159159…. & = \frac{53}{333} \end{matrix}$
0.3232……
$\begin{matrix} 0.323232… & = \frac{32}{100} + \frac{32}{10000} + …. \\ D o n d e : \\ a = \frac{32}{100}, r = \frac{\frac{32}{10000}}{\frac{32}{100}} = \frac{1}{100} \\ S & = \frac{a}{1 - r} \\ S & = \frac{\frac{32}{100}}{1 - \frac{1}{100}} \\ S & = \frac{\frac{32}{100}}{\frac{99}{100}} \\ S & = \frac{32}{99} \\ 0.323232…. & = \frac{32}{99} \end{matrix}$
0.144144…….
$\begin{matrix} 0.144144… & = \frac{144}{1000} + \frac{144}{1000000} + …. \\ D o n d e : \\ a = \frac{144}{1000}, r = \frac{\frac{144}{1000000}}{\frac{144}{1000}} = \frac{1}{1000} \\ S & = \frac{a}{1 - r} \\ S & = \frac{\frac{144}{1000}}{1 - \frac{1}{1000}} \\ S & = \frac{\frac{}{1000}}{\frac{}{1000}} \\ S & = \frac{48}{333} \\ 0.144144144…. & = \frac{48}{333} \end{matrix}$
0.3555……
$\begin{matrix} 0.35555… & = \frac{3}{10} + \frac{5}{100} + \frac{5}{1000} + \frac{5}{10000} …. \\ D o n d e : \\ a = \frac{5}{100}, r = \frac{\frac{5}{1000}}{\frac{5}{100}} = \frac{1}{10} \\ S & = \frac{a}{1 - r} \\ S & = \frac{\frac{5}{100}}{1 - \frac{1}{10}} \\ S & = \frac{\frac{5}{100}}{\frac{9}{10}} \\ S & = \frac{5}{90} \\ 0.35555…. & = \frac{3}{10} + \frac{5}{90} = \frac{27 + 5}{90} \\ 0.35555…. & = \frac{32}{90} \end{matrix}$
0.18111….
$\begin{matrix} 0.18111… & = \frac{18}{100} + \frac{1}{1000} + \frac{1}{10000} + …. \\ D o n d e : \\ a = \frac{1}{1000}, r = \frac{\frac{1}{10000}}{\frac{1}{1000}} = \frac{1}{10} \\ S & = \frac{a}{1 - r} \\ S & = \frac{\frac{1}{1000}}{1 - \frac{1}{10}} \\ S & = \frac{\frac{1}{1000}}{\frac{9}{10}} \\ S & = \frac{1}{900} \\ 0.1811111…. & = \frac{18}{100} + \frac{1}{900} = \frac{162 + 1}{900} \\ 0.1811111…. & = \frac{163}{900} \end{matrix}$
0.31818….
$\begin{matrix} 0.3181818… & = \frac{3}{10} + \frac{18}{1000} + \frac{18}{100000} + …. \\ D o n d e : \\ a = \frac{18}{1000}, r = \frac{\frac{18}{100000}}{\frac{18}{1000}} = \frac{1}{100} \\ S & = \frac{a}{1 - r} \\ S & = \frac{\frac{18}{1000}}{1 - \frac{1}{100}} \\ S & = \frac{\frac{}{1000}}{\frac{}{100}} \\ S & = \frac{2}{110} = \frac{1}{55} \\ 0.3181818…. & = \frac{3}{10} + \frac{1}{55} = \frac{33 + 2}{110} = \frac{}{} \\ 0.3181818…. & = \frac{7}{22} \end{matrix}$
2.1818……
$\begin{matrix} 2.181818… & = 2 + \frac{18}{100} + \frac{18}{10000} + …. \\ D o n d e : \\ a = \frac{18}{100}, r = \frac{\frac{18}{10000}}{\frac{18}{100}} = \frac{1}{100} \\ S & = \frac{a}{1 - r} \\ S & = \frac{\frac{18}{100}}{1 - \frac{1}{100}} \\ S & = \frac{\frac{}{100}}{\frac{}{100}} \\ S & = \frac{2}{11} \\ 2.181818…. & = 2 + \frac{2}{11} = \frac{22 + 2}{11} \\ 2.181818…. & = \frac{24}{11} \end{matrix}$