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?

Answer 1

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.