2019 AMC 10B Problem 18

Henry decides one morning to do a workout, and he walks \tfrac{3}{4} of the way from his home to his gym. The gym is 2 kilometers away from Henry's home. At that point, he changes his mind and walks \tfrac{3}{4} of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks \tfrac{3}{4} of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked \tfrac{3}{4} of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point A kilometers from home and a point B kilometers from home. What is |A-B|?

\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 1\frac{1}{5} \qquad \textbf{(D) } 1\frac{1}{4} \qquad \textbf{(E) } 1\frac{1}{2}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.