Question

5- Considerando a soma 0,5 + 0,05 + 0,005 + 0,0005 +,..., resolva. Qual a dízima periódica que representa esta soma. Determine a fração geratriz da dízima periódica que você escreveu no item a.​

Answer 1

$\boxed{\begin{array}{l}\sf 0,5+0,05+0,005+0,0005+...\\\sf a_1=0,5=\dfrac{5}{10}\\\sf a_2=0,05=\dfrac{5}{100}\\\sf q=\dfrac{a_2}{a_1}\\\\\sf q=\dfrac{\frac{5}{100}}{\frac{5}{10}}\\\sf q=\dfrac{\backslash\!\!\!5}{10\diagup\!\!\!0}\cdot\dfrac{1\diagup\!\!\!0}{\backslash\!\!\!5}=\dfrac{1}{10}\end{array}}$

$\boxed{\begin{array}{l}\underline{\rm soma~dos~termos~da~PG~infinita}\\\sf S_n=\dfrac{a_1}{1-q}\\\\\sf S_n=\dfrac{\frac{5}{10}}{1-\frac{1}{10}}\\\\\sf S_n=\dfrac{\frac{5}{10}}{\frac{9}{10}}\\\\\sf S_n=\dfrac{5}{\diagdown\!\!\!\!\!\!10}\cdot\dfrac{\diagdown\!\!\!\!\!\!10}{9}\\\\\sf S_n=\dfrac{5}{9}\end{array}}$