Question

dizima periodica composta 7,1321321...

Answer 1

Temos o seguinte:

$7,1321321...$

Podemos reescrever da seguinte maneira:

$7 + 0,1321321...= 7 + \frac{132}{1000} + \frac{132}{1000000} +... = 7 + \sum^{\infty}_{n=1} \frac{132}{1000^n}$

Resolvendo a somatória, com soma infinita:

$q = \frac{1}{1000} \\ \\ S_{n} = \frac{a_{1}}{1 - q} \therefore S_{n} = \dfrac{ \frac{132}{1000} }{1 - \frac{1}{1000} } = \frac{132}{1000} \cdot \frac{1000}{999 } = \frac{132}{999} = \frac{44}{333}$

Logo temos:

$7,1321321... = 7 + \frac{44}{333} = \frac{2331 + 44}{333} = \boxed{ \frac{2375}{333} }$