a yellow sqaure of 1 is divided into 9 smaller sqaures, and the middle square i colored blue. Each of the smaller yellow sqaures, is in turn divided into 9 sqaures, and each middle sqaure is colored blue. If this process continues indefinitely, what is the total area colored blue?
Let me ask you the same question:
If the distance to a wall is 1 meter, and in each step you go 1/9 th the way remaining. If you do this an infinite number of times, how far will you have traveled?
1/9+ 8/9 * 1/9 + 1/9*(8/9)^2+ ...
1/9(1+ (8/9)^2 + (8/9)^3 + ..)
or 1/9 the sum of the geometric progression a=1, r= 8/9
You can use the sum formula for an infinite geometric series here.
To solve this problem, we can analyze the pattern and find a general formula to calculate the total area colored blue.
Let's start with the first division. The yellow square of side length 1 is divided into 9 smaller squares, and the middle square is colored blue. So, the area colored blue in the first division is 1/9.
In the second division, each of the smaller yellow squares (with side length 1/3) is divided into 9 squares, and the middle squares are colored blue. Therefore, the area colored blue in the second division is (1/3)^2 * 1/9 = 1/81.
Continuing this pattern, we can see that in each division, the area colored blue reduces by a factor of 1/9. Therefore, the formula to calculate the area colored blue in the nth division is given by:
(1/9)^n
We are interested in the total area colored blue as the division continues indefinitely. In other words, we want to find the limit as n approaches infinity:
Limit[(1/9)^n] as n → infinity
Since (1/9)^n approaches 0 as n gets larger, the total area colored blue as the division continues indefinitely is 0.
Hence, the total area colored blue is 0.
To find the total area colored blue, we need to determine the area of the blue square in each iteration and sum them up.
Let's look at the pattern:
In the first iteration, we have a yellow square of size 1. The middle square is colored blue, so its area is 1 x 1 = 1.
In the second iteration, each of the 9 smaller yellow squares is divided into 9 squares. Since the area of the original yellow square is 1, each smaller yellow square has an area of 1/9. The middle square in each smaller yellow square is colored blue, so its area is (1/9) x (1/9) = 1/81.
In each subsequent iteration, the area of the blue square decreases by a factor of 9 because each square is divided into 9 smaller squares. Therefore, in the third iteration, the blue square area is (1/81) x (1/9) = 1/729.
This pattern continues indefinitely, and we can see that the area of the blue square in each iteration is decreasing by a factor of 9.
To find the total area colored blue, we need to sum up the areas of the blue squares in each iteration. This can be represented as:
1 + 1/81 + 1/729 + ...
This is a geometric series with a common ratio of 1/81. The formula to find the sum of an infinite geometric series is:
Sum = a / (1 - r),
where a is the first term and r is the common ratio.
In our case, a = 1 and r = 1/81. Plugging these values into the formula, we have:
Sum = 1 / (1 - 1/81) = 1 / (80/81) = 81/80.
Therefore, the total area colored blue is 81/80.