Question

Transforme em fração as geratrizes
a)0,2323.... b)1,666.. c)0,2222

Answer 1

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$\Large\boxed{\sf{\underline{Soma~dos~termos~de~uma~PG~infinita}}}\\\huge\boxed{\boxed{\boxed{\boxed{\boxed{\sf S_n=\dfrac{a_1}{1-q}}}}}}$

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$\tt{a)}~\sf{\underbrace{0,2323...=0,23...+0,0023+0,000023+...}_{soma~dos~termos~da~P.G~infinita}}$

$\sf{a_1=\dfrac{23}{100}}\\\sf{a_2=\dfrac{23}{10000}}$

$\sf{q=\dfrac{\frac{23}{10000}}{\frac{23}{100}}}\\\sf{q=\dfrac{\diagup\!\!\!2\diagdown\!\!\!3}{100\diagup\!\!\!0\diagup\!\!\!0}\cdot\dfrac{1\diagup\!\!\!0\diagup\!\!\!0}{\diagup\!\!\!2\diagdown\!\!\!3}\implies q=\dfrac{1}{100}}$

$\sf{S_n=\dfrac{\frac{23}{100}}{1-\frac{1}{100}}}\\\sf{S_n=\dfrac{\frac{23}{100}}{\frac{99}{100}}}\\\sf{S_n=\dfrac{23}{\diagup\!\!\!\!\!\!1\diagdown\!\!\!\!\!00}\cdot\dfrac{\diagup\!\!\!\!\!\!1\diagdown\!\!\!\!\!\!00}{99}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{S_n=\dfrac{23}{99}}}}}}$

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$\tt{b)}~\sf{1,666...=1+\underbrace{0,66...+0,0066...+0,00066...}_{soma~dos~termos~da~P.G~infinita}}\\\sf{a_1=0,66=\dfrac{66}{100}}\\\sf{a_2=\dfrac{66}{10000}}\\\sf{q=\dfrac{a_2}{a_1}}$

$\sf{q=\dfrac{a_2}{a_1}}\\\sf{q=\dfrac{\frac{66}{10000}}{\frac{66}{100</p><p>}}$

$\sf{q=\dfrac{\diagup\!\!\!6\diagdown\!\!\!6}{100\diagdown\!\!\!\!\!\!\diagup\!\!\!00}\cdot\dfrac{\diagup\!\!\!\!\!\!1\diagdown\!\!\!\!\!\100}{\diagup\!\!\!6\diagdown\!\!\!6}}\\\sf{q=\dfrac{1}{100}}$

$\sf{S_n=\dfrac{\frac{66}{100}}{1-\frac{1}{100}}}\\\sf{S_n=\dfrac{\frac{66}{100}}{\frac{99}{100}}}\\\sf{S_n=\dfrac{\diagdown\!\!\!66^{2}}{\diagup\!\!\!\!\!\!1\diagdown\!\!\!\!\!00}\cdot\dfrac{\diagup\!\!\!\!1\diagdown\!\!\!\!\!\000}{\diagdown\!\!\!\!\!99_{3}}}\\\sf{S_n=\dfrac{2}{3}}$

$\sf{1,666...=1+0,666...=1+\dfrac{2}{3}}\\\huge\boxed{\boxed{\boxed{\boxed{\boxed{\sf{1,666...=\dfrac{5}{3}}}}}}}$

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$\tt{c)}~\sf{\underbrace{0,222....=0,2+0,02...+0,002}_{soma~dos~termos~da~P.G~infinita}}\\\sf{a_1=0,2=\frac{2}{10}}\\\sf{a_2=0,02=\frac{2}{100}}\\\sf{q=\dfrac{a_2}{a_1}}\\\sf{q=\dfrac{\frac{2}{100}}{\frac{2}{10}}}$

$\sf{q=\dfrac{\diagdown\!\!\!2}{10\diagup\!\!\!\!0}\cdot\dfrac{1\diagup\!\!\!\!0}{\diagdown\!\!\!2}\implies q=\dfrac{1}{10}}$

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$\ell ife=\displaystyle\sf{\int_{birth}^{death}\dfrac{happiness}{time}dtime}$