The sides of the outside square in the figure are 8 inches long. The corners of the next smaller square are at the midpoints of the outside square, and so on.

Question

a.) find the sum of the perimeters of the first three squares.

Imagine that the process of nesting squares continues forever. The sum of the perimeters approaches a finite number.
b.) What is that number?

Answer 1

clearly the ratio of each smaller square is 1/√2. So, you have a geometric progression with

a = 8
r = 1/√2

Now just apply your formulas for the sum of such a sequence.