Question

Each week Matt washes 3 more cars than the week before. He started washing 6 cars his first week. Write an explicit formula that can be used to find the number of cars washed on any given day
An=6+3n
An=3+6(n-1)
An=6(3n-1)
An=6+3(n-1)

Answer 1

To derive the explicit formula for the number of cars washed by Matt in any given week \( n \), we start with the information provided:

1. In the first week (\( n = 1 \)), Matt washes \( A_1 = 6 \) cars.
2. Each subsequent week, he washes 3 more cars than the previous week.

So, for any week \( n \), the number of cars washed can be represented as:

- First week: \( A_1 = 6 \)
- Second week: \( A_2 = 6 + 3 \times 1 = 9 \)
- Third week: \( A_3 = 6 + 3 \times 2 = 12 \)
- Fourth week: \( A_4 = 6 + 3 \times 3 = 15 \)

From this pattern, we can create a general formula. The number of cars washed in week \( n \) can be expressed as:

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

Thus, we can see that:

- When \( n = 1 \), \( A_1 = 6 + 3(1 - 1) = 6 \)
- When \( n = 2 \), \( A_2 = 6 + 3(2 - 1) = 9 \)
- When \( n = 3 \), \( A_3 = 6 + 3(3 - 1) = 12 \)
- When \( n = 4 \), \( A_4 = 6 + 3(4 - 1) = 15 \)

Now, comparing the choices given:

1. \( A_n = 6 + 3n \)
2. \( A_n = 3 + 6(n - 1) \)
3. \( A_n = 6(3n - 1) \)
4. \( A_n = 6 + 3(n - 1) \)

The correct explicit formula for the number of cars washed in week \( n \) is:

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

So the answer is **\( A_n = 6 + 3(n - 1) \)**.