# Sum of infinite geometric series

A regular hexagon is inscribed in a circle of radius 100. The area of that hexagon outside the square is shaded. Another circle is inscribed in the hexagon, and hexagon is inscribed in the second circle. The area of the second circle which lies outside the hexagon is shaded. This process continues to infinity.

What is the sum of all shaded areas?

Radius of 1 = 100

Radius of 2 = ______

Radius of 3 = _______

Radius of 4 = ________

Side of hexagon 1st =_________

Side of hexagon 2nd = ________

Side of hexagon 3rd = ________

Side of hexagon 4th = _______

Area of Circle 1 = ________

Area of Circle 2 = ________

Area of Crcle 3 =________

Area of Circle 4 = _______

Area of Hexagon 1 = _______

Area of Hexagon 2 =______

Area of Hexagon 3 = ______

Area of Hexagon 4 = _________

Difference of 1st =_________

Difference of 2nd =_________

Difference of 3rd =_________

Difference of 4th =________

Each new one is _________of the previous area.

Sum of all areas = ___________

Is there any relationship concerning the areas as the number of sides of the inscribed figure increases? If so what would you predict would be the area sum for 8 or 9 sides?_________

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#### Solution Preview

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The sum of an infinite Geometric Series

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A regular hexagon is inscribed in a circle of radius 100. The area of that hexagon outside the square is shaded. Another circle is inscribed in the hexagon, and hexagon is inscribed in the second circle. The area of the second circle which lies outside the hexagon is shaded. This ...

#### Solution Summary

This is an infinite geometric series question using the hexagons and circles inscribed inside of each other.

Geometric Series and Sums

Please see the attached Microsoft Word document.

Thanks for your expertise.

Part A Find a specific geometric that sums up to 30.

Part B Can you find a specific geometric series that sums to -30 ?

Part C Can you find a specific geometric series summing up to - ?

Part D Determine precisely which real numbers can be sums of a geometric series.