Half of halves

Half of the square we cut off, then half of the rest, etc. Five cuts we made in this way.

What part of the content of the original square is the content of the cut part?

Result

r =  0.969

Solution:

$q=1/2=\dfrac{ 1 }{ 2 }=0.5 \ \\ S=1 \ \\ S_{1}=S - q \cdot \ S=1 - 0.5 \cdot \ 1=\dfrac{ 1 }{ 2 }=0.5 \ \\ S_{2}=S_{1} - q \cdot \ S_{1}=0.5 - 0.5 \cdot \ 0.5=\dfrac{ 1 }{ 4 }=0.25 \ \\ S_{3}=S_{2} - q \cdot \ S_{2}=0.25 - 0.5 \cdot \ 0.25=\dfrac{ 1 }{ 8 }=0.125 \ \\ S_{4}=S_{3} - q \cdot \ S_{3}=0.125 - 0.5 \cdot \ 0.125=\dfrac{ 1 }{ 16 }=0.0625 \ \\ S_{5}=S_{4} - q \cdot \ S_{4}=0.0625 - 0.5 \cdot \ 0.0625 \doteq \dfrac{ 1 }{ 32 } \doteq 0.0313 \ \\ \ \\ r=S-S_{5}=1-0.0313 \doteq \dfrac{ 31 }{ 32 } \doteq 0.9688 \doteq 0.969$

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