From the description, we can see a pattern in the number of people seated at the tables:
- 1 table seats 4 people
- 2 tables seat 6 people (3 people per table)
- 3 tables seat 8 people (2.67 people per table)
To analyze the pattern mathematically:
- 1 Table: 4 people \( \rightarrow 1 \text{ table} \)
- 2 Tables: 6 people \( \rightarrow 3 \text{ people per table} \)
- 3 Tables: 8 people \( \rightarrow \frac{8}{3} \approx 2.67 \text{ people per table} \)
This suggests a series with the following pattern:
- For the first table, there are 4 people - this seems to be a fixed amount.
- For every additional table, it seems the number of seats increases.
Let's express \( n \) as the number of tables:
- For 1 table: \( 4 \times 1 = 4 \)
- For 2 tables: \( 3 \times 2 = 6 \)
- For 3 tables: \( 2.67 \times 3 \approx 8 \)
If we denote the total number of people as \( P \):
- When we generalize the number of people with the increment established in the progression, we can find a formula \( P(n) \).
Continuing this pattern, we can project forward:
- If we set up the equation based on the last known quantities:
- For \( n = 4 \): It should seat \( 4 + 2 = 10 \)
- For \( n = 5 \): It should seat \( 4 + 3 = 12 \)
- For \( n = 6 \): It should seat \( 4 + 4 = 14 \)
- For \( n = 7 \): It should seat \( 4 + 5 = 16 \)
Thus for 16 people, we find that 7 tables would be needed.