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Artyom0805 [142]
3 years ago
7

Find the geometric sum: 40 + 20 + 10 + … + 0.078125

Mathematics
1 answer:
sveticcg [70]3 years ago
4 0
S=40+20+10+\cdots+0.078125
S=40\left(1+\dfrac12+\dfrac14+\cdots+\dfrac1{512}\right)
S=40\left(1+2^{-1}+2^{-2}+\cdots+2^{-9}\right)

2^{-1}S=40\left(2^{-1}+2^{-2}+2^{-3}+\cdots+2^{-10}\right)

\implies S-2^{-1}S=40\left(1-2^{-10}\right)
\implies\dfrac12S=40\left(1-\dfrac1{1024}\right)
\implies S=80\left(1-\dfrac1{1024}\right)=\dfrac{5115}{64}=79.921875
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