8. Problem 8 A square with area 84 is divided into 4 squares as shown in the diagram to the right. The upper left square is colored black. The lower right square is again divided into 4 squares and we repeat the procedure forever. The process is repeated an infinite number of times. What is the total area that is colored black? For this problem, we first find the area of the big black square in the top left. That is equal to 1 fourth of the total area of the square. So 1 fourth times 84 equals 21. The second biggest black square which is in the top left corner of the square in the bottom right is equal to 1 fourth of 1 fourth of 84. So 1 fourth times 21 or 21 over 4. And actually this is a geometric series, each time the area of the black square divides by 4 and it becomes 1 fourth of the previous term. So the first term is 21, then 21 over 4, then the next term would be 21 over 16, all the way for infinity. This is a geometric series where the first term a is equal to 21 and the common ratio between the terms is equal to 1 fourth. And if you recall, for an infinite geometric series where the common ratio has an absolute value less than 1, for example 1 fourth, we can actually find the sum using the formula a over 1 minus r. Plugging our numbers into this formula, we get 21 over 1 minus 1 fourth equals 21 times 4 thirds equals 28. So the correct answer is b.

infinite geometric series

square division

black area calculation

geometric series sum

area of squares