Let's determine the explicit formula that describes Marika's training regimen for her race.
1. Understand the Initial Situation:
- Marika begins by sprinting an initial distance of 100 yards.
2. Understand the Increment:
- Each day, she adds 4 yards to her sprinting distance.
3. Identify the Number of Days:
- She trains for a total of 21 days.
4. Arithmetic Sequence:
- The situation can be modeled by an arithmetic sequence because Marika increases her sprinting distance by a fixed amount each day.
- In an arithmetic sequence, the n-th term can be calculated using the formula:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
where:
- [tex]\( a_n \)[/tex] is the distance on day n,
- [tex]\( a_1 \)[/tex] is the initial distance,
- [tex]\( n \)[/tex] is the term number (in this context, the day),
- [tex]\( d \)[/tex] is the daily increase (common difference).
5. Set the Values:
- The initial distance ([tex]\( a_1 \)[/tex]) is 100 yards.
- The daily increase ([tex]\( d \)[/tex]) is 4 yards.
- We need to find the explicit formula to calculate the distance on any day [tex]\( n \)[/tex].
6. Plug in the Values:
- Using the arithmetic sequence formula:
[tex]\[
a_n = 100 + (n - 1) \cdot 4
\][/tex]
7. Compare with the Given Choices:
- A. [tex]\( a_n = 100 + (n - 1) \cdot 4 \)[/tex]
- B. [tex]\( a_n = 21 + (n - 1) \cdot 4 \)[/tex]
- C. [tex]\( a_n = 100 + (n - 1) \cdot 21 \)[/tex]
- D. [tex]\( a_n = 4 + (n - 1) \cdot 100 \)[/tex]
The correct formula matches choice A.
8. Correct Choice:
- Therefore, the explicit formula that models Marika’s training is:
[tex]\[
\boxed{a_n = 100 + (n-1) \cdot 4}
\][/tex]
The correct answer is A.
