What is the sum of the perimeters of all the squares?

Question

The perimeter of the largest square is 13 inches. The perimeter of

each subsequent square is 75% of the perimeter of the previous

square.  Assume you can continue this design indefinitely.  

What is the sum of the perimeters of all the squares.

Answer 1

The answer is a geometric progression: 13+3/4*13+(3/4)^2*13+... The nth term of this is 13*(3/4)^n. Call the sum of this progression S. 3/4S= 13*3/4+13*(3/4)^2+13*(3/4)^3+...+13*(3/4)^(n+1), so subtracting 3/4S from S we get: S/4=13(1-(3/4)^(n+1)). When n gets very large (approaches infinity), (3/4)^(n+1) approaches zero. Therefore S=4*13=52. The total perimeter then converges to 52 inches, assuming the squares do not have common boundaries.