english

# Start with a square of side length 1. Join the midpoints of its four sides to form a smaller square

Start with a square of side length 1. Join the midpoints of its four sides to form a smaller square and 4 right triangles. Let A1 be the area of one of the right triangles. Now, join the midpoints of each side of the smaller square to form a new even smaller square and 4 new right triangles. Let A2 be the area of one of the new right triangles. Continue the process forever.

Requried:

a. Find a formula for A.

b. Determine whether the sequence A converges. If it does converge, what is the limit?