Explicit Formula

Question

Matt has started a car washing business to save money for college. Each week, he washes 3 more cars than he did the week before. Matt started by washing 6 cars his first week in business. Write an explicit formula that can be used to find the number of cars he washed on any given week.

Answer 1

To find an explicit formula for the number of cars Matt washes in any given week, we can identify the pattern in the number of cars he washes each week.

1. Identify the First Week's Value:

   • In the first week, Matt washes 6 cars.

2. Identify the Increase per Week:

   • Each week, he washes 3 more cars than the previous week.

3. Pattern Recognition:

   • Week 1: 6 cars
   • Week 2: 6 + 3 = 9 cars
   • Week 3: 9 + 3 = 12 cars
   • Week 4: 12 + 3 = 15 cars
   • And so on...

4. General Formula:

   • We can see this is an arithmetic sequence where:
     • The first term (number of cars washed in week 1) \( a_1 = 6 \)
     • The common difference (increase in cars washed per week) \( d = 3 \)

   The formula for the \( n \)-th term of an arithmetic sequence can be expressed as: \[ a_n = a_1 + (n-1) \cdot d \]

5. Plugging in the Values: \[ a_n = 6 + (n-1) \cdot 3 \]

6. Simplifying: \[ a_n = 6 + 3(n - 1) \] \[ a_n = 6 + 3n - 3 \] \[ a_n = 3n + 3 \]

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

\[ \boxed{3n + 3} \]

Answer 2

A
n
=
6
+
3
n
A
n
​
=6+3n

A
n
=
6
(
3
n
−
1
)
A
n
​
=6(3n−1)

A
n
=
6
+
3
(
n
−
1
)
A
n
​
=6+3(n−1)

A
n
=
3
+
6
(
n
−
1
)
A
n
​
=3+6(n−1)

Answer 3

Let's analyze the different formulas you've provided.

1. An = 6 + 3n:

   • This formula represents the number of cars washed in week \(n\), with a starting point and growth that doesn't align with the given pattern. It increases linearly but does not accurately represent the starting value and increment because it suggests that in week 1, Matt washes 9 cars.

2. An = 6(3n - 1):

   • This formula complicates things unnecessarily and does not correctly represent the arithmetic sequence.

3. An = 6 + 3(n - 1):

   • This formula correctly computes the number of cars washed in week \(n\). Simplifying this gives: \[ A_n = 6 + 3n - 3 = 3n + 3 \] This is equivalent to our original formula and is correct.

4. An = 3 + 6(n - 1):

   • This formula would indicate Matt washes only \(3\) cars in week 1, which is inconsistent with the information given (he washes 6 cars the first week). This scaling by \(6\) also seems unnecessary and does not reveal the proper growth pattern established.

Correct Explicit Formula

The correct explicit formula for the number of cars washed in week \(n\) is: \[ A_n = 3n + 3 \] or alternatively: \[ A_n = 6 + 3(n - 1) \] Either form is valid, with the version \(6 + 3(n - 1)\) emphasizing the starting point and growth.

If you have any further questions or need additional clarifications, feel free to ask!