Problem 101 - Precalculus: Mathematics for Calculus

Question

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 that is colored blue?

Step-by-Step Solution

Answer

The total area colored blue is 1.

Step 1: Initial Understanding

We start with a yellow square of side length 1, giving it an area of 1. It is divided into nine smaller squares, each with equal size.

Step 2: First Iteration

In the first iteration, the middle square is colored blue. As the entire big square is divided into 9 equal parts, the area of one small square is \( \frac{1}{9} \). Thus, the area of the blue square is initially \( \frac{1}{9} \).

Step 3: Second Iteration

Each of the 8 yellow squares left from the first division is subdivided into 9 smaller squares and their center square is colored blue. Therefore, each center blue square now has an area of \( \frac{1}{9^2} \), and the total area colored blue at this step is \( 8 \times \frac{1}{9^2} \).

Step 4: General Pattern and Series

Recognize that each iteration results in a series where the number of squares colored blue increases by a factor of 8, and each square has an area reduced by a factor of \( \frac{1}{9} \). Therefore, the total blue area can be represented by the infinite geometric series: \[ \frac{1}{9} + \frac{8}{9^2} + \frac{8^2}{9^3} + \cdots \]

Step 5: Calculating the Infinite Series

The series \( \frac{1}{9} + \frac{8}{9^2} + \frac{8^2}{9^3} + \cdots \) is a geometric series with first term \( a = \frac{1}{9} \) and common ratio \( r = \frac{8}{9} \). The sum of this infinite series is given by the formula \( S = \frac{a}{1 - r} \), so \[ S = \frac{\frac{1}{9}}{1 - \frac{8}{9}} = 1. \]

Step 6: Conclusion

Thus, as the process continues indefinitely, the total area colored blue converges to 1.

Key Concepts

Geometric Series

A geometric series is a type of infinite series with each term being a constant multiple of the previous term. In the context of our exercise, we begin with a square divided into smaller squares, where each center square is colored blue. This pattern continues indefinitely. To understand the series, note that:

• The first blue area is a single square with an area of \( \frac{1}{9} \).
• The next iteration adds more blue squares, each with an area reduced by a factor of \( \frac{1}{9} \) and there are 8 of these squares.
• This sequence of areas can be expressed as \( \frac{1}{9} + \frac{8}{9^2} + \frac{8^2}{9^3} + \cdots \).

Each term is essentially \( \frac{8^n}{9^{n+1}} \) where \( n \) is the iteration number. Recognizing this as a geometric series allows us to compute the total area of blue squares as the series progresses.

Area Calculation

Understanding how to calculate the total blue area involves identifying how the area of each new set of squares decreases. Initially, the large square has an area of 1, and the first blue square has an area of \( \frac{1}{9} \). Each subsequent step involves:

• Dividing each non-blue square into 9 smaller squares.
• Coloring its center square blue.

The key is to understand that every new blue square's area is \( \frac{1}{9} \) times the area of the yellow square it comes from. Therefore, as the process goes on:

• In the second generation, the new blue squares each have an area of \( \frac{1}{9^2} \).
• The total area for this generation of blue squares is \( 8 \times \frac{1}{9^2} \).

This pattern continues, forming the geometric series explained in the previous section.

Convergence

Convergence is a crucial concept when dealing with infinite series. In our exercise, the infinite series represents the total blue area, which converges to a specific value as more iterations are added. Understanding convergence helps us grasp that even though we're adding an infinite number of terms, the sum approaches a finite value.
The series we have, \( \frac{1}{9} + \frac{8}{9^2} + \frac{8^2}{9^3} + \cdots \), has a common ratio of \( \frac{8}{9} \), which is less than 1, ensuring convergence. The sum of this infinite series is calculated using the formula for an infinite geometric series:

• \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.
• In this case: \( S = \frac{\frac{1}{9}}{1 - \frac{8}{9}} = 1 \).

This means that, despite the infinite repetition of the process, the total blue area becomes exactly 1, which is the entire area of the original square.