Question

5) Transforme cada dízima periódica em fração irredutível.
a) 0,7777...
b) 0,353535...​

Answer 1

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$\tt{a)}~\sf{0,777...=\underbrace{0,7...+0,07...+0,007... +... }_{soma~dos~termos~da~PG~infinita}}\\\sf{a_1=\frac{7}{10}}\\\sf{a_2=\frac{7}{100}}\\\sf{q=\frac{a_2}{a_1}}\\\sf{q=\dfrac{\frac{\diagup\!\!\!7}{10\diagdown\!\!\!0}}{\frac{\diagup\!\!\!7}{1\diagdown\!\!\!0}}=\dfrac{1}{10}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{S_n=\dfrac{a_1}{1-q}}}}}}$

$\sf{S_n=\dfrac{\frac{7}{10}}{1-\frac{1}{10}}}\\\sf{S_n=\dfrac{\frac{7}{\diagdown\!\!\!\!\!\!10}}{\frac{9}{\diagdown\!\!\!\!\!\!10}}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{S_n=\dfrac{7}{9}\checkmark}}}}}$

$\tt{b)}~\sf{\underbrace{0,353535...=0,35+0,0035+0,000035...+...}_{Soma~dos~termos~de~uma~PG~infinita}}\\\sf{a_1=0,35=\frac{35}{100}}\\\sf{a_2=0,0035=\frac{35}{10~000}}\\\Huge\sf{q=\frac{\frac{\diagdown\!\!\!\!\!\!35}{100\diagdown\!\!\!\!\!\!00}}{\frac{\diagdown\!\!\!\!\!\!35}{1\diagdown\!\!\!\!\!\!00}}=\dfrac{1}{100}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{s_n=\dfrac{a_1}{1-q}}}}}}$

$\sf{S_n=\dfrac{\frac{35}{100}}{1-\frac{1}{100}}}\\\sf{S_n=\dfrac{\frac{35}{\diagup\!\!\!1\diagup\!\!\!0\diagup\!\!\!0}}{\frac{99}{\diagup\!\!\!1\diagup\!\!\!0\diagup\!\!\!0}}}\\\huge\boxed{\boxed{\boxed{\boxed{\tt{S_n=\dfrac{35}{99}\checkmark}}}}}$