Question

1. Utilize a fórmula da soma dos termos da P.G. infinita e mostre que a soma
1/2+1/4+1/16+1/32+1/64 ... é igual a 1
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Answer 1

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Soma dos termos de uma PG infinita

$\huge\boxed{\boxed{\boxed{\boxed{\sf S_n=\dfrac{a_1}{1-q}}}}}$

$\sf\underbrace{\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{32}+\dfrac{1}{64}+...}_{soma~dos~termos~de~uma~PG~infinita}\\\sf a_1=\dfrac{1}{2}=2^{-1}\\\sf a_2=\dfrac{1}{4}=2^{-2}\\\sf q=\dfrac{a_2}{a_1}\\\sf q=\dfrac{2^{-2}}{2^{-1}}=2^{-2+1}=2^{-1}=\dfrac{1}{2}$

$\sf S_n=\dfrac{a_1}{1-q}\\\sdf S_n=\dfrac{\frac{1}{2}}{1-\frac{1}{2}}\\\sf S_n=\dfrac{\frac{1}{2}}{\frac{1}{2}}\\\sf S_n=1\\\large\boxed{\boxed{\boxed{\boxed{\boxed{\sf\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{32}+\dfrac{1}{64}+...=1\checkmark}}}}}$