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Geometric Series : Infinite Series of Circles inside Squares and Equilateral Triangles

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A circle of radius 100is inscribed in a square. The inscribing process continues to infinity. What is the sum of the unshaded areas?
Radius of 1 = 100 - Radius of 2 = _________Radius of 3 = _______ Radius of 4 = ____________

Side of square 1 =________, Side of square 2 = __________, Side of square 3 = ___________Side of square 4 = ______________

Area of Circle 1 = ___________, Area of Circle 2 = _________, Area of Circle 3 = _____________Area of Circle 4 = _______________________
Area of Square 1 =_____________Area of Square 2 = ___________Area of Square 3=___________Area of Square 4 = _____________
Difference of 1st -_____________Difference of 2nd -________Difference of 3rd - ___________Difference of 4th - _______________

The next is same as above, but the circle with radius 100 is inscribed in an equilateral traiangle. The inscribing process continues until infinity. What is the sum of the unshaded area? All the above for this problem same as above.

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Solution Summary

A recursive expression is found for the area outside a square but inside a circles for an infinite set of squares and circles. A recursive expression is found for the area outside a triangle but inside a circles for an infinite set of triangles and circles. The solution is detailed and well presented. Diagrams are included.

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A circle of radius 100is inscribed in a square. The inscribing process continues to infinity. What is the sum of the unshaded areas?

See a graph above. For two circles and squares, we have the relation between the side x and radius r. Obviously, for the square and its inscribed circle, x=2r.

For two circles, their radius have a relation as follows.

Radius of 1 ...

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  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
  • "Thank you"
  • "Thank you very much for your valuable time and assistance!"
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