(UPF) A região em cinza do quadrado ABCD se repete infinitamer te, de acordo com o padráo representado na figura originando sempre mais quadrados. square square essa maneira , a fração do quadrado ABC D cará preenchida e (1)/(2) C) (1)/(3) e) 1(1)/(2)

## Step 1: Understanding the Pattern<br />### The image describes a fractal pattern where each square is divided into smaller squares, and some of these are shaded. This pattern repeats infinitely, suggesting a geometric series.<br /><br />## Step 2: Identifying the Shaded Area<br />### The largest inner square (top-left) is shaded, occupying $\frac{1}{4}$ of the total area of the large square ABCD. The second largest shaded square (bottom-right) also occupies $\frac{1}{4}$ of the remaining area, which is $\frac{1}{16}$ of the total area of ABCD. This pattern continues with each subsequent shaded square occupying $\frac{1}{4}$ of the remaining unshaded area.<br /><br />## Step 3: Calculating the Total Shaded Fraction<br />### The total shaded area forms an infinite geometric series with the first term $a = \frac{1}{4}$ and common ratio $r = \frac{1}{4}$. The sum of an infinite geometric series is given by the formula:<br /><br />\[<br />S = \frac{a}{1 - r}<br />\]<br /><br />Substituting the values, we get:<br /><br />\[<br />S = \frac{\frac{1}{4}}{1 - \frac{1}{4}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}<br />\]<br /><br />This means that $\frac{1}{3}$ of the total area of the square ABCD will be filled with shaded squares as the pattern continues infinitely.