Question

Qual a fração geratriz da dízima 0,3333?​

Answer 1

$\Large\boxed{\begin{array}{l}\sf x=0,333...\cdot10\\\sf 10x=3,333...\\\sf 10x=3+0,333...\\\sf10x=3+x\\\sf 10x-x=3\\\sf 9x=3\\\sf x=\dfrac{3\div3}{9\div3}\\\\\sf x=\dfrac{1}{3}\end{array}}$

$\boxed{\begin{array}{l}\underline{\rm Usando~a~soma}\\\underline{\rm dos~termos~da~PG~infinita}\\\huge\boxed{\boxed{\boxed{\boxed{\sf S_n=\dfrac{a_1}{1-q}}}}}\\\sf 0,333...=\underbrace{\sf 0,3...+0,03....+0,003...}_{\sf soma~dos~termos~da~PG~infinita}\\\\\sf a_1=0,3=\dfrac{3}{10}~~ a_2=0,03=\dfrac{3}{100}\\\\\sf q=\dfrac{a_2}{a_1}=\dfrac{\frac{3}{100}}{\frac{3}{10}}=\dfrac{\backslash\!\!\!3}{10\diagup\!\!\!0}\cdot\dfrac{1\diagup\!\!\!0}{\backslash\!\!\!3}=\dfrac{1}{10}\end{array}}$

$\Large\boxed{\begin{array}{l}\sf S_n=\dfrac{a_1}{1-q}\\\\\sf S_n=\dfrac{\frac{1}{10}}{1-\frac{1}{10}}=\dfrac{\frac{1}{10}}{\frac{10-1}{10}}\\\\\sf S_n=\dfrac{\frac{1}{10}}{\frac{9}{10}}=\dfrac{1}{\diagdown\!\!\!\!\!\!10}\cdot\dfrac{\diagdown\!\!\!\!\!\!10}{9}=\dfrac{1}{9}\end{array}}$

Answer 2

Resposta:

0,111...=1/9

0,22...=2/9

0,333...=3/9

0,444...=4/9

0,5555...=5/9

0,666....=6/9

0,777...=7/9

0,888...=8/9

0,99999...=9/9 =1