Question

Se x = 0,1212..., o valor numérico da expressão: é:

a) 1

/37

b) 21/37

c) 33/37

d) 43/37

e) 51/37​

Answer 1

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$\Large\boxed{\sf{\underline{Soma~dos~termos~da~P.G~infinita}}}\\\huge\boxed{\boxed{\boxed{\boxed{\boxed{\sf S_n=\dfrac{a_1}{1-q}}}}}}$

$\sf\underbrace{0,1212...=0,12...+0,0012...+0,000012...+...}_{soma~dos~termos~da~P.G~infinita}\\\sf a_1=0,12=\frac{12}{100}\\\sf a_2=0,0012=\frac{12}{10000}\\\sf q=\dfrac{a_2}{a_1}\\\sf q=\dfrac{\frac{12}{10000}}{\frac{12}{100}}$

$\sf q=\dfrac{\diagdown\!\!\!\!\!\!12}{100\diagup\!\!\!0\diagup\!\!\!0}\cdot\dfrac{1\diagup\!\!\!0\diagup\!\!\!0}{\diagdown\!\!\!\!\!\!\!12}=\dfrac{1}{100}$

$\sf S_n=\dfrac{\frac{12}{100}}{1-\dfrac{1}{100}}\\\sf S_n=\dfrac{\frac{12}{100}}{\frac{100-1}{100}}\\\sf S_n=\dfrac{\frac{12}{100}}{\frac{99}{100}}\\\sf S_n=\dfrac{12}{1\diagup\!\!\!0\diagup\!\!\!0}\cdot\dfrac{1\diagup\!\!\!0\diagup\!\!\!0}{99}\\\sf S_n=\dfrac{12\div3}{99\div3}$

$\huge\boxed{\boxed{\boxed{\boxed{\sf S_n=\dfrac{4}{33}}}}}$