if the sheding process show in the figure is continued indef

if the sheding process show in the figure is continued indefinitely, what fractional part of the largest square will eventually be shaded?

Solution

Largest square has 1/4rth shaded

Then 1/4rth of 1/4rth of largest square = (1/4)(1/4)

Then 1/4 rth of 1/4rth second largest of 1/4rth of largest = (1/4)(1/4)(1/4

and so on...

Let largest square has area a sq units:

So, Shaded area = a/4 +a/16 + a/64 ------ infinity

= a[ 1/4 +1/16 +1/64 ------- inf]

= a/4 [ 1+ 1/4 +1/6 -------- inf]

Its a G.P. series with common ratio = 1/4

Sum of G.P. series = 1/(1-r)

total shaded area = a/4[ 1/(1-1/4) ]

= a/4[ 1/3/4]

=a/4(4/3)

= a/3

So, 1/3 area of the largest square is shaded