Question
Each hour before a large sporting event, there are 2 times as many seats filled in the stadium as there were the previous hour. If you make a list of the total number of seats filled each hour, what kind of sequence will you have?
Answers
GPT-4o mini
The situation you described can be modeled as a geometric sequence.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor. Here, you mentioned that there are 2 times as many seats filled each hour as there were the previous hour. This means that if we let \( a_0 \) be the number of seats filled at the starting hour (let's say hour 0), then the number of seats filled at hour 1 would be \( 2a_0 \), at hour 2 would be \( 4a_0 \) (which is \( 2^2 a_0 \)), at hour 3 would be \( 8a_0 \) (which is \( 2^3 a_0 \)), and so on.
In general, the number of seats filled at hour \( n \) can be expressed as:
\[
a_n = 2^n a_0
\]
where \( n \) is the hour number. Since \( n \) varies, the total number of seats filled forms a geometric sequence with a common ratio of 2 (specifically, each term is multiplied by 2 to get the next term).
Thus, the answer is that you will have a **geometric sequence**.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor. Here, you mentioned that there are 2 times as many seats filled each hour as there were the previous hour. This means that if we let \( a_0 \) be the number of seats filled at the starting hour (let's say hour 0), then the number of seats filled at hour 1 would be \( 2a_0 \), at hour 2 would be \( 4a_0 \) (which is \( 2^2 a_0 \)), at hour 3 would be \( 8a_0 \) (which is \( 2^3 a_0 \)), and so on.
In general, the number of seats filled at hour \( n \) can be expressed as:
\[
a_n = 2^n a_0
\]
where \( n \) is the hour number. Since \( n \) varies, the total number of seats filled forms a geometric sequence with a common ratio of 2 (specifically, each term is multiplied by 2 to get the next term).
Thus, the answer is that you will have a **geometric sequence**.