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

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.

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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