Question

The cost of having one shirt laundered is $6. Each additional shirt costs $4.

What is the explicit formula to represent the situation?
What is the recursive formula to represent the situation?

Answer 1

To represent the situation of laundering shirts with a cost structure, we can define explicit and recursive formulas.

1. **Explicit Formula**:
Let \( n \) represent the number of shirts laundered. The cost for laundering the first shirt is $6, and each additional shirt costs $4. Therefore, the total cost \( C(n) \) for laundering \( n \) shirts can be expressed as:

\[
C(n) =
\begin{cases}
6 & \text{if } n = 1 \\
6 + 4(n - 1) & \text{if } n > 1
\end{cases}
\]

This can also be simplified into a single expression for all \( n \geq 1 \):

\[
C(n) = 6 + 4(n - 1) = 4n + 2
\]

where \( n \geq 1 \).

2. **Recursive Formula**:
To formulate a recursive definition, we define the cost based on the number of shirts.

- Let \( C(1) = 6 \) be the base case (the cost of laundering one shirt).
- For \( n > 1 \), the cost of laundering \( n \) shirts can be represented using the cost of \( (n - 1) \) shirts:

\[
C(n) = C(n-1) + 4
\]

This recursive formula means that the cost to launder \( n \) shirts is the cost to launder \( (n - 1) \) shirts plus $4 for the additional shirt.

In summary:
- **Explicit Formula**: \( C(n) = 4n + 2 \) for \( n \geq 1 \)
- **Recursive Formula**:
- \( C(1) = 6 \)
- \( C(n) = C(n-1) + 4 \) for \( n > 1 \)