The side of the large square is 1 unit and each black point divides its corresponding segment in half. If this blue area continues in this pattern, then what is the total area in blue?
So the largest blue square is 1/4 of the total.
The second largest is 1/4 of 1/4 of the total = 1/16
The third is (1/4)^3 = 1/64.
The fourth is (1/4)^4 = 1/256
The 2 fifths are 2* (1/4)^5 = 2*1/1024 = 1/512
We add these up giving
I'm not sure this is correct but pretty sure it is.
if you don't understand anything feel free to ask!