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

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 2

Not right

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: \[ a_n = 3n + 3 \]

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