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=12=0.5 S=1 S1=Sq S=10.5 1=12=0.5 S2=S1q S1=0.50.5 0.5=14=0.25 S3=S2q S2=0.250.5 0.25=18=0.125 S4=S3q S3=0.1250.5 0.125=116=0.0625 S5=S4q S4=0.06250.5 0.06251320.0313  r=SS5=10.031331320.96880.969q=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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