1. (UPF) A região em cinza do quadrado ABCD se repete infinitame nte, de acordo com o padrão representado na figura , originando sempre mais quadrados. square square square Dessa maneira , a fração do quadrado ABC D que ficará preenchida é a) (1)/(2) c) (1)/(3) e) 13 b) (1)/(4) d) (5)/(4)

## Step 1: Understand the Pattern<br />### The pattern described in the image is similar to a Fibonacci tiling, where each square's side length follows the Fibonacci sequence. In such a pattern, the shaded regions represent squares whose areas are part of this sequence.<br /><br />## Step 2: Calculate the Total Area of Shaded Squares<br />### If the largest square (bottom left) has a side length of $F_n$, then the next smaller square (top left) has a side length of $F_{n-1}$, and so on. The total area of these squares can be expressed as the sum of the areas of the individual squares: $F_n^2 + F_{n-1}^2 + \ldots + F_1^2$.<br /><br />## Step 3: Use the Property of Fibonacci Sequence<br />### A known property of the Fibonacci sequence is that the sum of the squares of the first $n$ Fibonacci numbers is equal to the product of the $n$th and $(n+1)$th Fibonacci numbers: $F_1^2 + F_2^2 + \ldots + F_n^2 = F_n \cdot F_{n+1}$.<br /><br />## Step 4: Determine the Fraction of the Large Square<br />### Since the entire large square ABCD is composed of these Fibonacci squares, the fraction of the square that is filled by the shaded regions is determined by the ratio of the sum of the areas of the shaded squares to the area of the large square. As the pattern repeats infinitely, it approaches a limit.<br /><br />## Step 5: Conclude the Fraction<br />### The infinite repetition of this pattern means that the shaded area will eventually fill half of the large square ABCD. This is because the sum of the areas of the shaded squares converges to half the area of the large square.