If the blue triangles continue to be formed indefinitely, what fraction of the big square will be shaded blue?
+128408 Call the side of the square, S
Area of largest blue triangle = (S)^2 / 2 = S^2 /2
Area of next largest blue triangle = (S/2)^2 / 2 = S^2 / 8
This infinite series continues as S^2 /32 ,etc. (each successive square has 1/4 the area of the previous one = the common ratio )
So the sum of the areas is S^2 (1/2 + 1/8 + 1/32 + ....... )
The sum of the series inside the parentheses is given as first term /( 1 - common ratio )
(1/2) / ( 1 -1/4) = (1/2) / (3/4) = (1/2) (4/3) = 4 / 6 = 2/3
So
The fraction of the square covered = 2/3
