(UPF) A região em cinza do quadrado ABCD se repete infinitament e, de acordo com o padrão representa do na figura ,originando sempre mais quadrados. square square square square Dessa maneira, a fração do quadrado A BCD que ficará preenchid e a) (1)/(2) C) (1)/(3) e) 1(1)/(2)

## Step 1: Understanding the Pattern<br />### The large square ABCD is divided into smaller squares, with a fractal-like pattern where grey and white squares alternate. The pattern repeats infinitely, creating more squares within each iteration.<br /><br />## Step 2: Calculating the Fraction of Grey Area<br />### To determine the fraction of the square that will be shaded grey, we need to analyze the pattern's repetition and how much area each iteration adds to the grey region.<br />### Initially, the bottom-left quarter of the square is grey. In each subsequent iteration, smaller grey squares are added in a similar pattern.<br /><br />## Step 3: Summing the Series<br />### The initial grey area is $\frac{1}{4}$ of the total area. Each subsequent iteration adds grey squares that are $\frac{1}{4}$ of the remaining white area from the previous iteration.<br />### This forms an infinite geometric series where the first term $a = \frac{1}{4}$ and the common ratio $r = \frac{1}{4}$.<br /><br />\[<br />\text{Sum} = a + ar + ar^2 + ar^3 + \ldots = \frac{a}{1 - r}<br />\]<br /><br />### Substituting the values:<br /><br />\[<br />\text{Sum} = \frac{\frac{1}{4}}{1 - \frac{1}{4}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}<br />\]<br /><br />## Step 4: Conclusion<br />### Therefore, the fraction of the square ABCD that will be filled with grey squares is $\frac{1}{3}$.