Alright, let's analyze the provided payment schedule and derive the explicit formula to calculate your pay for any given week.
1. Understand the Pattern:
- Week 1: \[tex]$11.00
- Week 2: \$[/tex]11.50
- Week 3: \[tex]$12.00
- Week 4: \$[/tex]12.50
Upon examining the payments, we observe an increasing pattern.
2. Determine the Increment:
Calculate the difference in payment between consecutive weeks:
[tex]\[
\text{Increment} = \text{Pay in Week 2} - \text{Pay in Week 1} = \$11.50 - \$11.00 = \$0.50
\][/tex]
Therefore, the pay increases by \[tex]$0.50 each week.
3. Establish the Initial Payment:
The initial payment (week 1) is \$[/tex]11.00.
4. Create the General Formula:
We can express the payment for any week [tex]\( n \)[/tex] using the initial payment and the weekly increment. The general form of an arithmetic sequence is:
[tex]\[
\text{Pay(n)} = \text{Initial Payment} + (n - 1) \times \text{Increment}
\][/tex]
For our specific case:
[tex]\[
\text{Pay(n)} = 11.00 + (n - 1) \times 0.50
\][/tex]
5. Explicit Formula:
Therefore, the explicit formula to calculate the payment for any week [tex]\( n \)[/tex] is:
[tex]\[
\text{Pay(n)} = 11.00 + (n - 1) \times 0.50
\][/tex]
### Example Usage:
To illustrate how to use this formula, let's calculate the pay for week 10:
[tex]\[
\text{Pay(10)} = 11.00 + (10 - 1) \times 0.50
\][/tex]
[tex]\[
\text{Pay(10)} = 11.00 + 9 \times 0.50
\][/tex]
[tex]\[
\text{Pay(10)} = 11.00 + 4.50
\][/tex]
[tex]\[
\text{Pay(10)} = 15.50
\][/tex]
Thus, according to our explicit formula, the pay for week 10 is \$15.50.
