Question

Transforme as dizimas periódicas abaixo em frações geratrizes

a) 0,755...
b) 0,3488888...​

Answer 1

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Soma dos termos de uma PG infinita

$\Large\boxed{\boxed{\boxed{\boxed{\sf S_n=\dfrac{a_1}{1-q}}}}}$

$\tt a)~\sf 0,755...=0,7+\underbrace{0,05...+0,005...+0,00005+...}_{soma~dos~termos~da~PG~infinita}\\\sf a_1=0,05=\frac{5}{100}\\\sf a_2=0,005=\frac{5}{1000}\\\sf q=\dfrac{a_2}{a_1}$

$\sf q=\dfrac{\frac{5}{1000}}{\frac{5}{100}}\\\sf q=\dfrac{\backslash\!\!\!5}{10\diagdown\!\!\!\!\!00}\cdot\dfrac{1\diagdown\!\!\!\!\!00}{\backslash\!\!\!5}=\dfrac{1}{10}\\\sf S_n=\dfrac{\frac{5}{100}}{1-\frac{1}{10}}\\\sf S_n=\dfrac{\frac{5}{100}}{\frac{9}{10}}\\\sf S_n=\dfrac{5}{10\backslash\!\!\!0}\cdot\dfrac{1\backslash\!\!\!0}{9}=\dfrac{5\div5}{90\div5}=\dfrac{1}{18}$

$\sf 0,7555...=0,7+\dfrac{1}{18}=\dfrac{7}{10}+\dfrac{1}{18}\\\sf =\dfrac{63+5}{90}=\dfrac{68\div2}{90\div2}=\dfrac{34}{45}$

$\tt b)~\sf 0,34888...=0,34+\underbrace{0,008...+0,0008...+...}_{soma~dos~termos~da~PG~infinita}\\\sf a_1=0,008=\frac{8}{1000}\\\sf a_2=0,0008=\frac{8}{10000}\\\sf q=\dfrac{a_2}{a_1}\\\sf q=\dfrac{\frac{8}{10000}}{\frac{8}{1000}}$

$\sf q=\dfrac{\diagup\!\!\!8}{10\bf{\diagdown\!\!\!\!\!\!00}\diagup\!\!\!0}\cdot\dfrac{1\diagup\!\!\!\!\!00\diagup\!\!\!0}{\diagup\!\!\!8}=\dfrac{1}{10}$

$\sf S_n=\dfrac{\frac{8}{1000}}{1-\frac{1}{10}}\\\sf S_n=\dfrac{\frac{8}{1000}}{\frac{9}{10}}\\\sf S_n=\dfrac{8}{100\backslash\!\!\!0}\cdot\dfrac{1\backslash\!\!\!0}{9}=\dfrac{8\div4}{900\div4}=\dfrac{2}{225}$

$\sf 0,34888...=0,34+\dfrac{2}{225}=\dfrac{34\div2}{100\div2}+\dfrac{2}{225}\\\sf =\dfrac{17}{50}+\dfrac{2}{225}\\\sf 0,34888...=\dfrac{153+4}{450}=\dfrac{157}{450}$