Half the people on a bus get off at each stop after the first, and no one gets on after the first stop. If exactly one person gets off at stop number 7, how many people got on the bus at the first stop?

Introduction / Context:
This is a logical and numerical reasoning problem involving repeated halving. At every stop after the first, half the people on the bus get off. Given that one person gets off at the seventh stop, you have to work backwards to find how many people originally boarded the bus at the first stop. This type of question tests your ability to handle geometric progressions and reverse reasoning in word problems.

Given Data / Assumptions:

• All passengers get on the bus at the first stop.
• After the first stop, no new passengers get on the bus.
• At each stop after the first, exactly half of the people on the bus get off.
• At stop number 7, exactly one person gets off.
• We must find how many people originally boarded at the first stop.

Concept / Approach:
If at each stop half the people get off, then the number of people getting off follows a pattern where at each stop the number of passengers just before the stop is double the number who get off at that stop. At stop 7, one person gets off, so there must have been 2 people on the bus just before that stop. By repeatedly doubling the number of people backwards for each previous stop, we can find the number of passengers just before stop 2, which is the same as the number of people who got on at the first stop (assuming the bus was empty before that). This is effectively a geometric sequence with common ratio 2 when tracing backwards.

Step-by-Step Solution:
Step 1: At stop 7, one person gets off. Since half the people get off at each stop, the number of people on the bus just before stop 7 must have been 2. Step 2: Before stop 6, the number of people must have been double the number just before stop 7, because half get off at stop 6 and the remaining half move on. So before stop 6, there were 4 people. Step 3: Similarly, before stop 5 there were 8 people, before stop 4 there were 16, before stop 3 there were 32, and before stop 2 there were 64 people on the bus. Step 4: The people on the bus before stop 2 are exactly the people who boarded at the first stop (since no one gets on after the first stop and we assume the bus was empty before). Step 5: Therefore, the number of people who got on the bus at the first stop is 64.

Verification / Alternative check:
We can verify by moving forward. Start with 64 people. At stop 2, half (32) get off, leaving 32. At stop 3, half (16) get off, leaving 16. At stop 4, half (8) get off, leaving 8. At stop 5, half (4) get off, leaving 4. At stop 6, half (2) get off, leaving 2. At stop 7, half (1) get off, leaving 1 person on the bus. This matches the condition that exactly one person gets off at stop 7, so the reasoning is consistent.

Why Other Options Are Wrong:
Option 128: If 128 people boarded, then at stop 7 the number getting off would be 2, not 1.
Option 32: Working forwards with 32 passengers gives 0.5 person at some stage, which is impossible.
Option 16: Similar to 32, this would lead to a fractional passenger count when halved repeatedly.
Option 256: This would lead to 4 people getting off at stop 7, not 1.

Common Pitfalls:
The main mistake is to try to follow the process forward instead of backward, which can be confusing and often leads to miscounting. Another common error is to subtract half instead of correctly halving the current number at each step. Always remember that if half of the people get off, the number remaining is also half, and that backward reasoning (doubling) is often easier when you know the final condition.

Final Answer:
The number of people who got on at the first stop is 64.