Infinite series

9000062909

Level: 
B
Consider the square of the side \(4\, \mathrm{cm}\). 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.
\(32 + 16\sqrt{2}\)
\(32 - 16\sqrt{2}\)
\(32\)
\(\infty \)

9000062910

Level: 
B
Consider the square of the side \(4\, \mathrm{cm}\). 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 squares of all squares.
\(32\)
\(40\)
\(\frac{32} {3} \)
\(\infty \)

9000073401

Level: 
B
In the following list identify the expression which equals \(3.3\overline{12}\).
\(3.3 +\sum _{ n=1}^{\infty }12\cdot 10^{-2n-1}\)
\(3 +\sum _{ n=1}^{\infty }312\cdot 10^{-2n-1}\)
\(3 +\sum _{ n=1}^{\infty }312\cdot 10^{-3n}\)
\(3.3 +\sum _{ n=1}^{\infty }12\cdot 10^{-3n}\)

9000073402

Level: 
B
In the following list identify the expression which equals \(- 1.0\overline{345}\).
\(- 1 -\sum _{n=1}^{\infty }345\cdot 10^{-3n-1}\)
\(- 1 -\sum _{n=1}^{\infty }345\cdot 10^{-3n}\)
\(-\sum _{n=1}^{\infty }(10 + 345\cdot 10^{-3n-1})\)
\(1 -\sum _{n=1}^{\infty }345\cdot 10^{-3n}\)