3psilon
  • 3psilon
Dad dreamed that he put $1 on the first square,$2 on the second, $4 dollars on the third, and so on, doubling the amount each time.If it costs $65,535 to cover all the squares,how many squares were in Dad's dream?
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
\[1+2+2^2+2^3+...+2^{n-1}=2^n-65535\]
anonymous
  • anonymous
oops i mean \[=2^n-1=65535\]
3psilon
  • 3psilon
My book says there are 16 squares but only 1 covered ?

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anonymous
  • anonymous
yes there are 16
anonymous
  • anonymous
\[2^n-1=65535\] \[2^n=65526\] and \[2^{16}=65536\]
anonymous
  • anonymous
another typo, \(2^n=65536\)
anonymous
  • anonymous
It's a geometric series, right? So in general we use: \[ \Large \sum_{i=0}^{n-1}a_0r^i = a_0\frac{1-r^n}{1-r} \]
anonymous
  • anonymous
Where \(a_0 = 1\) and \(r=2\)...
anonymous
  • anonymous
65535 = 2^16 - 1 --> 16 squares

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