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

Question

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

Answer 1

## Step 1: Understand the Structure of the Diagram### The diagram is a large square ABCD divided into four equal smaller squares. Three of these quadrants (lower left, lower right, and upper left) are fully occupied by one square each. The upper right quadrant is further subdivided into a pattern of nested squares.## Step 2: Analyze the Pattern in the Upper Right Quadrant### The upper right quadrant contains a recursive pattern where each subsequent square is smaller than the previous one. This pattern continues infinitely, creating an infinite series of progressively smaller squares.## Step 3: Calculate the Area of the Shaded Region### Since the pattern repeats infinitely, we need to calculate the total area covered by these nested squares as a fraction of the entire square ABCD. Each time a new square is added, it occupies a fraction of the remaining space in the upper right quadrant.## Step 4: Determine the Fraction of the Entire Square### The shaded region in the upper right quadrant represents a geometric series. The sum of this infinite series can be calculated using the formula for the sum of an infinite geometric series:

, where

is the first term and

is the common ratio.### In this case, the first term

is

(since the largest square in the upper right quadrant is

of the entire square), and the common ratio

is also

because each subsequent square is

of the previous one.## Step 5: Calculate the Sum of the Series### Using the formula, the sum of the series is

.## Step 6: Determine the Total Shaded Area### The total shaded area is the sum of the areas of the three full quadrants plus the sum of the infinite series in the upper right quadrant. Therefore, the total shaded area is

.## Step 7: Simplify the Expression### To add these fractions, find a common denominator:

and

. Adding these gives

.

Question

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

Solution

Resposta

Explicação