A square is divided into nine smaller squares of equal area. The center square is then divided into nine smaller squares of equal area and the pattern continues indefinitely. What fractional part of the figure is shaded?

Guest May 10, 2021

#4**0 **

4/9 + 4/81 + 4/729 + … = (4/9)/(1-1/9) = (4/9)/(8/9) = (4/9)(9/8) = 4/8 = 1/2

MathProblemSolver101 May 10, 2021

#5**+1 **

I think it is an infinite geometric sequence with a1 = 4/9 r = 1/9

sum = a1 / (1-r) = 4/8 = 1/2

ElectricPavlov May 10, 2021

#6**+1 **

You can think of the square without the center piece. This figure would consist of \(8\) squares, \(4\) being shaded or \(\frac{1}{2}\) shaded. We can think of the center piece of this figure as \(8\) squares again with \(4\) shaded. Each time you count the center piece with \(8\) squares, you have \(\frac{1}{2}\) shaded and \(\frac{1}{2}\) unshaded. This pattern continues forever. Thus the answer is \(\frac{1}{2}.\)

Qyper May 10, 2021