Question
Market researchers are studying the effects of sending an advertisement through text messaging. On the first day of the advertisement program, the researcher sent a text message to 8 people. On the next day, each of those people will send the text message to another 8 people, and so on. The pattern of sending the advertisement through text messaging is shown in the table.
Number of Days Number of People Receiving Text Message
1 8
2 8×8
3 8×8×8
4 8×8×8×8
Number of Days Number of People Receiving Text Message
1 8
2 8×8
3 8×8×8
4 8×8×8×8
Answers
GPT-4o mini
The pattern you have described follows a geometric sequence where the number of people receiving the text message increases exponentially each day.
Here's the breakdown for the number of people receiving the text message each day:
- On Day 1, the number of people receiving the text message is \(8\).
- On Day 2, each of those 8 people sends the message to 8 new people. So, the total for Day 2 is \(8 \times 8 = 8^2\).
- On Day 3, those \(8^2\) people will each send the message to 8 new people, resulting in \(8 \times 8 \times 8 = 8^3\).
- On Day 4, the number of recipients will be \(8^4\) as the same process continues.
The general formula for the number of people receiving the text message on Day \(n\) is:
\[
\text{Number of People on Day } n = 8^n
\]
Using this formula, we can calculate the number of people receiving the text message for any day \(n\):
- For Day 1: \(8^1 = 8\)
- For Day 2: \(8^2 = 64\)
- For Day 3: \(8^3 = 512\)
- For Day 4: \(8^4 = 4096\)
Thus, the number of people receiving the text message increases exponentially.
Here's the breakdown for the number of people receiving the text message each day:
- On Day 1, the number of people receiving the text message is \(8\).
- On Day 2, each of those 8 people sends the message to 8 new people. So, the total for Day 2 is \(8 \times 8 = 8^2\).
- On Day 3, those \(8^2\) people will each send the message to 8 new people, resulting in \(8 \times 8 \times 8 = 8^3\).
- On Day 4, the number of recipients will be \(8^4\) as the same process continues.
The general formula for the number of people receiving the text message on Day \(n\) is:
\[
\text{Number of People on Day } n = 8^n
\]
Using this formula, we can calculate the number of people receiving the text message for any day \(n\):
- For Day 1: \(8^1 = 8\)
- For Day 2: \(8^2 = 64\)
- For Day 3: \(8^3 = 512\)
- For Day 4: \(8^4 = 4096\)
Thus, the number of people receiving the text message increases exponentially.