Problem 77 - Algebra and Trigonometry

Question

Geometry A yellow square of side 1 is divided into nine smaller squares, and the middle square is colored blue as shown in the figure. Each of the smaller yellow squares is in turn divided into nine squares, and each middle square is colored blue. If this process is continued indefinitely, what is the total area colored blue?

Step-by-Step Solution

Answer

The total area colored blue is 1.

Step 1: Analyze the Initial Division

The original yellow square has a side of 1, giving it an area of 1 as well. Dividing it into 9 smaller equal squares means each smaller square has side length \(\frac{1}{3}\) and an area of \(\frac{1}{9}\). Thus, the initial middle blue square has an area of \(\frac{1}{9}\).

Step 2: Further Subdivision Process

Each yellow square continues the subdivision process. Every yellow square is subdivided into 9 smaller squares, with each middle square becoming blue. The side length for these second-level squares is \(\frac{1}{3^2}\), and their areas are \(\frac{1}{9^2}\).

Step 3: Calculate Area of New Blue Squares at Each Level

At every stage of division, there are 8 new yellow squares that each create one blue square. The area of one blue square in level \(n\) is \(\left(\frac{1}{3^n}\right)^2 = \frac{1}{9^n}\), and there are \(8^{(n-1)}\) new blue squares at each level after the initial division.

Step 4: Set Up the Infinite Series for the Blue Area

The total blue area is the sum of an infinite series: \[\frac{1}{9} + 8 \cdot \frac{1}{9^2} + 8^2 \cdot \frac{1}{9^3} + \cdots\]. This is a geometric series where the first term \(a = \frac{1}{9}\) and the common ratio \(r = \frac{8}{9}\).

Step 5: Find the Sum of the Infinite Geometric Series

The sum \(S\) of an infinite geometric series with first term \(a\) and common ratio \(r\) (where \(r < 1\)) is given by \(S = \frac{a}{1-r}\). Substituting the values, we have: \[S = \frac{\frac{1}{9}}{1 - \frac{8}{9}} = \frac{\frac{1}{9}}{\frac{1}{9}} = 1\].

Key Concepts

Area Calculation

In geometry, the concept of calculating areas is crucial for understanding the properties of shapes. When starting with a larger shape and breaking it down into smaller shapes, knowing how to calculate the area of each part becomes essential.
In the original exercise, we start with a large yellow square with a total area of 1. By dividing it into 9 smaller squares, the area of each smaller square becomes \(\frac{1}{9}\), since the square is evenly divided.
This simple division helps us understand how the area changes as we continue the subdivision process. Calculating the area of the initial blue square involves simply recognizing that it also has an area of \(\frac{1}{9}\) because it is one of those smaller squares. This straightforward calculation sets the stage for more complex subdivisions and area calculations in later steps.

Subdivision Process

The concept of subdivision is important for understanding complex patterns and figures. In this exercise, the subdivision process starts with a single large square and divides it into smaller squares repeatedly.
The first step of subdivision divides the large square into 9 smaller squares, each having a side length of \(\frac{1}{3}\). As this process continues, each yellow square becomes a new larger square in the next step.
Each of these new squares is also divided into 9 smaller squares. The middle square of each subdivision becomes blue, representing a critical part of this geometric progression. As you can see:

• The subdivision starts with a single large square.
• Each subdivision produces smaller squares with side lengths shrinking by a factor of 3 each time.
• The middle square of each subdivision becomes the focus as it is colored blue and contributes to the total blue area.

Understanding this process is crucial for visualizing how a simple starting figure can transform into a complex pattern through repeated subdivisions.

Geometric Progression

A geometric progression (or series) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this context, the geometric progression arises from the repeated subdivision and coloring of squares.
The infinite series in this exercise comes into play when we want to calculate the total area covered in blue. As the number of subdivisions approaches infinity, an infinite number of blue squares appear, each smaller than the last. However, they follow a predictable pattern:
The initial blue square has an area of \(\frac{1}{9}\). As subdivisions proceed, each subsequent level's blue squares get progressively smaller and more numerous.
The sum total of areas forms an infinite geometric series:

• First term \(a = \frac{1}{9}\)
• Common ratio \(r = \frac{8}{9}\), since each level includes 8 times more new blue squares as before, with each square having an area \(\frac{1}{9}\) of the previous level.

The formula for the sum of an infinite geometric series \(S = \frac{a}{1-r}\) helps us find the total area colored blue. In this case, substituting the values gives \(S = 1\), meaning that if this process continues indefinitely, the entire original area will be filled with blue squares.