Half of halves

We cut half of the square off, then half of the rest, etc. Five cuts we made in this way. What part of the area of the original square is the area of the cut part?

Final Answer:

r =  31/32

Step-by-step explanation:

$$\begin{gathered} q=1/2=\frac{1}{2}=0.5 \\ S=1 \\ {S}_1=S-q\cdot S=1-0.5\cdot 1=\frac{1}{2}=0.5 \\ {S}_2=S_1-q\cdot S_1=0.5-0.5\cdot 0.5=\frac{1}{4}=0.25 \\ {S}_3=S_2-q\cdot S_2=0.25-0.5\cdot 0.25=\frac{1}{8}=0.125 \\ {S}_4=S_3-q\cdot S_3=0.125-0.5\cdot 0.125=\frac{1}{16}=0.0625 \\ {S}_5=S_4-q\cdot S_4=0.0625-0.5\cdot 0.0625=\frac{1}{32}\doteq 0.0313 \\ r=S-S_5=1-0.0313=\frac{31}{32}=0.9688 \end{gathered}$$

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