Question

Use the image to answer the question.

An illustration shows a square with four stars along the edges, two squares attached with six stars along the outer edges; and three squares attached with eight stars along the outer edges.

The tables for a party are seated as 4 people at 1 table, 6 people at 2 tables, and 8 people at 3 tables. Use a sequence to determine how many tables would be needed for 16 people.

Answer 1

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. **1 Table**: 4 people \( \rightarrow 1 \text{ table} \)
2. **2 Tables**: 6 people \( \rightarrow 3 \text{ people per table} \)
3. **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.