[tex]\large\boxed{\begin{array}{l}\sf0,222... =0,2+0,02... +0,002...\\\sf a_1=0,2=\dfrac{2}{10}~~a_2=0,02=\dfrac{2}{100}\\\sf q=\dfrac{a_2}{a_1}\\\sf q=\dfrac{\frac{2}{100}}{\frac{2}{10}}\\\sf q=\dfrac{\diagup\!\!\!2}{10\backslash\!\!\!0}\cdot\dfrac{1\backslash\!\!\!0}{\diagup\!\!\!2}=\dfrac{1}{10}\\\sf S_n=\dfrac{a_1}{1-q}\\\sf S_n=\dfrac{\frac{2}{10}}{1-\frac{1}{10}}\\\sf S_n=\dfrac{\frac{2}{10}}{\frac{9}{10}}\\\sf S_n=\dfrac{2}{\diagdown\!\!\!\!\!\!10}\cdot\dfrac{\diagdown\!\!\!\!\!\!10}{9}\\\sf S_n=\dfrac{2}{9}\end{array}}[/tex]
