Question

Fração geratriz de 2,5656565656565...
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Answer 1

Temos o seguinte:

$2,565656... = 2 + 0,565656...$

Assim:

$0,565656... = \frac{56}{100} + \frac{56}{10000} + \frac{56}{1000000} = \sum\limits^{\infty}_{n = 1} \dfrac{56}{100^n}$

Logo realizando a soma infinita:

$S_{n} = \dfrac{a_{1}}{1 - q} , \text{sendo: } q = \dfrac{1}{100}$

$\sum\limits^{\infty}_{n = 1} \dfrac{56}{100^n} = \dfrac{ \frac{56}{100} }{1 - \frac{1}{100} } = \dfrac{ \frac{56}{100} }{ \frac{99}{100} } = \dfrac{56}{100} * \dfrac{100}{99} = \dfrac{56}{99}$

Logo:

$2,565656... = 2 + \sum\limits^{\infty}_{n = 1} \dfrac{56}{100^n} = 2 + \dfrac{56}{99} = \dfrac{99*2 + 56}{99} = \boxed{ \dfrac{254}{99} }$