A sheet of paper has an initial thickness of T units. It is folded in half 35 times, and each fold doubles the thickness of the sheet. After 35 such equal folds, what will be the new thickness of the paper?

Introduction / Context:
This question tests your understanding of exponential growth through repeated doubling. Each time the paper is folded, its thickness doubles. After multiple folds, the thickness grows very rapidly, and the resulting expression involves powers of 2 rather than simple multiplication by the number of folds.

Given Data / Assumptions:

• The initial thickness of the paper is T (in some unit of length).

• The paper is folded 35 times in half.

• Each fold doubles the thickness; that is, after one fold, the thickness becomes 2T, after two folds it becomes 4T, and so on.

• We assume idealised folding with no tearing or compression other than the mathematical doubling of thickness.

Concept / Approach:
When a quantity doubles repeatedly, we model it using powers of 2. After n doublings, the final amount is the original amount multiplied by 2^n. Here, because each fold is a doubling, the number of folds directly indicates the exponent of 2 in the expression for the final thickness.

Step-by-Step Solution:
Step 1: Before any fold, thickness = T. Step 2: After the 1st fold, thickness doubles: Thickness = T * 2. Step 3: After the 2nd fold, thickness doubles again: Thickness = T * 2 * 2 = T * 2^2. Step 4: After the 3rd fold, thickness doubles again: Thickness = T * 2^2 * 2 = T * 2^3. Step 5: Continuing this pattern, after n folds, thickness = T * 2^n. Step 6: In this problem, n = 35. Therefore, the final thickness after 35 folds is: Thickness = T * 2^35.

Verification / Alternative check:
You can check small values to confirm the rule. For 1 fold, thickness is T * 2^1. For 2 folds, T * 2^2. For 3 folds, T * 2^3. The exponent always matches the number of folds, so the pattern is correct. Extending it to 35 folds gives T * 2^35 naturally.

Why Other Options Are Wrong:
Option B (35T) treats the process as simple repeated addition instead of doubling; it would be correct if each fold added T, not doubled the thickness.
Option C (2 * 35 * T) is also linear in 35 and does not represent exponential growth; it simply multiplies T by 70.
Option D (T / 35) suggests that the paper becomes thinner with folding, which is the opposite of what actually happens.

Common Pitfalls:
A frequent error is confusing repeated doubling with repeated addition and writing expressions like T + 35T instead of T * 2^35. Another mistake is to mix up the exponent and the multiplier, e.g., writing T * 35^2. Always remember that doubling n times corresponds to multiplying by 2^n, not by 2n or n^2.

Final Answer:
Therefore, after 35 folds, the thickness of the paper will be T * 2^35.