Infinite series

9000062906

Level: 
B
A semicircle is a half of the full circle. An infinite spiral is built from semicircles with a decreasing radius. The radius of the first semicircle is 2cm. The radius of each semicircle in the spiral is one half of the radius of the previous semicircle. Find the total length of the spiral.
4π
43π
−4π
∞

9000062908

Level: 
B
A quarter circle is an arc formed by one quarter of the full circle. An infinite spiral is built from quarter circles with an increasing radius. The radius of the first quarter circle is 4cm. The radius of each quarter circle in the spiral is one half of the radius of the previous quarter circle. Find the total length of the spiral.
4π
8
83
∞

9000062909

Level: 
B
Consider the square of the side 4cm. The second square is inscribed into this first square by joining the centers of all sides. In a similar way, the third square is inscribed into the second square by joining the centers of the sides of the second square and this process continues up to infinity. Find the sum of the perimeters of all squares.
32+162
32−162
32
∞