1. (UPF) A região em cinza do quadrado ABCD se repete infinitame nte, de acordo com o padrão represent ado na figura , originando sempre mais quadrados. square square square square square square Dessa maneira, a fração do quadrado AB CD que ficará preenchida é a) (1)/(2) c) (1)/(3) e) 1(1)/(2)

## Step 1: Understanding the Pattern<br />### The image describes a large square divided into smaller squares, following a pattern resembling the Fibonacci sequence. This pattern involves recursively subdividing squares, where each subsequent square is smaller than the previous one.<br /><br />## Step 2: Identifying the Gray Area<br />### The gray areas in the image are part of a recursive process that continues infinitely. Each new gray square is smaller than the last, and they are arranged in a way that suggests they form a geometric series.<br /><br />## Step 3: Calculating the Fraction of the Gray Area<br />### To determine the fraction of the entire square that will be filled by the gray regions, we need to consider the sum of an infinite geometric series. If each gray square's area is a fraction of the previous one, the total area covered by the gray squares can be calculated using the formula for the sum of an infinite geometric series: $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio.<br /><br />## Step 4: Applying the Geometric Series Formula<br />### Assuming the largest gray square "A" covers half of the large square ABCD (since it occupies one quadrant), and each subsequent gray square is half the size of the previous one, the series would be: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots$. This is a geometric series with $a = \frac{1}{2}$ and $r = \frac{1}{2}$.<br /><br />## Step 5: Solving the Series<br />### Using the formula $S = \frac{a}{1 - r}$, we find:<br />\[<br />S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1<br />\]<br />This means the gray area will eventually fill the entire square ABCD.