Question

Matt washed 3 more cars each week than the week before. He washes 6 cars the first week. Write an explicit formula that can be used

Answer 1

Not right

Answer 2

To find an explicit formula for the number of cars Matt washes in week \( n \), we can recognize that the pattern describes an arithmetic sequence.

1. **First Week**: Matt washes 6 cars.
2. **Subsequent Weeks**: Matt washes 3 more cars each week than the week before.

This results in the following sequence for the number of cars washed:
- Week 1: \( 6 \)
- Week 2: \( 6 + 3 = 9 \)
- Week 3: \( 9 + 3 = 12 \)
- Week 4: \( 12 + 3 = 15 \)
- ...

The general term for an arithmetic sequence can be expressed as:

\[
a_n = a_1 + (n - 1) \cdot d
\]

where:
- \( a_n \) is the \( n \)-th term (the number of cars washed in week \( n \)),
- \( a_1 \) is the first term (6 cars),
- \( d \) is the common difference (3 cars),
- \( n \) is the week number.

Substituting the known values:

\[
a_n = 6 + (n - 1) \cdot 3
\]

Now, simplifying this formula:

\[
a_n = 6 + 3n - 3
\]
\[
a_n = 3n + 3
\]

Thus, the explicit formula for the number of cars Matt washes in week \( n \) is:

\[
a_n = 3n + 3
\]

Answer 3

I apologize for any confusion. Let's reevaluate the formulation based on the information given.

1. **First Week**: Matt washes **6** cars.
2. **After each week**: He washes **3** more cars than the previous week.

- Week 1: \( 6 \) cars.
- Week 2: \( 6 + 3 = 9 \) cars.
- Week 3: \( 9 + 3 = 12 \) cars.
- Week 4: \( 12 + 3 = 15 \) cars.

From this, it looks like we have:
- Week 1: \( 6 \)
- Week 2: \( 6 + 3 \)
- Week 3: \( 6 + 2(3) \)
- Week 4: \( 6 + 3(3) \)

So, the number of cars washed in week \( n \) can be defined as:
\[
a_n = 6 + (n - 1) \cdot 3
\]

Now, let's simplify:

\[
a_n = 6 + 3n - 3
\]
\[
a_n = 3n + 3
\]

It appears that the calculation led to the same formula again. To ensure clarity:
- \( a_1 = 6 \)
- \( a_2 = 9 \)
- \( a_3 = 12 \)
- ...

So the explicit formula for the number of cars washed in week \( n \) is indeed:
\[
a_n = 3n + 3
\]

If there's a specific part you would like to highlight or if there's further context, please let me know!