Are the areas of the two shieets of paper ever the same after each being folded \( n \) times? Explain how you know.

To determine if the areas of the two sheets of paper are ever the same after each being folded \( n \) times, we need to consider the process of folding and how it affects the area of the paper. When a sheet of paper is folded, the area of the paper is reduced by half with each fold. This is because the folded paper creates a new layer on top of the original paper, effectively halving the area. Let's consider the areas of the two sheets of paper after each fold: 1. After the first fold, the area of the first sheet of paper is reduced to half, and the area of the second sheet of paper is also reduced to half. 2. After the second fold, the area of the first sheet of paper is reduced to a quarter, and the area of the second sheet of paper is also reduced to a quarter. 3. After the third fold, the area of the first sheet of paper is reduced to an eighth, and the area of the second sheet of paper is also reduced to an eighth. From this pattern, we can see that after each fold, the areas of both sheets of paper are reduced by half. This means that the areas of the two sheets of paper are never the same after each being folded \( n \) times. Therefore, we know that the areas of the two sheets of paper are never the same after each being folded \( n \) times because the area of each sheet is reduced by half with each fold.