Question

Obtenha as frações geratrizes do número 0,3444444444...​

Answer 1

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$\Large\boxed{\sf{\underline{Soma~dos~termos~da~PG~infinita}}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{S_n=\dfrac{a_1}{1-q}}}}}}\\\boxed{\begin{array}{c}\sf{a_1\implies1^{\underline{o}}~termo}\\\sf{q\implies raz\tilde{a}o~da~progress\tilde{a}o}\end{array}}$

$\sf{0,3444...=0,3+\underbrace{0,04...+0,004....+0,0004+...}_{soma~dos~termos~da~PG~infinita}}\\\sf{a_1=0,04=\dfrac{4}{100}}\\\sf{a_2=0,004=\dfrac{4}{1000}$

$\sf{q=\dfrac{a_2}{a_1}}\\\sf{q=\dfrac{\frac{4}{1000}}{\frac{4}{100}}}\\\sf{q=\dfrac{\backslash\!\!\!4}{10\diagdown\!\!\!\!\!00}\cdot\dfrac{1\diagdown\!\!\!\!\!\!00}{\backslash\!\!\!4}=\dfrac{1}{10}}$

$\sf{S_n=\dfrac{\frac{4}{100}}{1-\frac{1}{10}}}\\\sf{S_n=\dfrac{\frac{4}{100}}{\frac{9}{10}}}\\\sf{S_n=\dfrac{4}{10\backslash\!\!\!0}\cdot\dfrac{1\backslash\!\!\!0}{9}=\dfrac{4}{90}}$

$\sf{0,3444...=\dfrac{3}{10}+\dfrac{4}{90}}\\\sf{0,3444...=\dfrac{27+4}{90}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{0,3444...=\dfrac{31}{90}}}}}}$