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February 16th, 2015, 10:29 AM   #1
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Can anyone help me? (inscribed squares)

Hi there! I am struggling with this problem, I am just not sure what to do!

I had an idea for what to do for letter a, but I'm not sure about it. And if you could help with b too, that would be great!
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February 16th, 2015, 10:57 AM   #2
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Each square (other than the first) has an area of 1/2 of the preceding square. See if you can figure out why.
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February 16th, 2015, 01:36 PM   #3
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If you can't think of an algebraic way to show that, consider what happens when you fold a square along the edges of the next one. Count the thicknesses of paper that make up each square.
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February 16th, 2015, 06:38 PM   #4
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Let $\displaystyle M$ be the midpoints of the sides of the square with sides $\displaystyle S_n$. Then the vertices of the square with sides $\displaystyle S_{n-1}$ are at $\displaystyle M$. By the Pythagorean theorem,

$\displaystyle \left(\frac12 S_n\right)^2+\left(\frac12 S_n\right)^2=S^2_{n-1}$

$\displaystyle \frac14 S^2_n+\frac14 S^2_n=S^2_{n-1}$

$\displaystyle \frac12 S^2_n=S^2_{n-1}$

Hence the general term is $\displaystyle \left(\frac12\right)^{n-1}$ and the first five terms are $\displaystyle 1,\frac12,\frac14,\frac18,\frac{1}{16}$.

Last edited by greg1313; February 16th, 2015 at 07:40 PM.
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