Sporis To train for an upcoming marathon, Olivia plans to run 3 miles per day for the first week and then increase the daily distance by a half mile each of the following weeks. a. Write an equation to represent the \( n \)th term of the sequence. b. If the pattern continues, during which week will she run 10 miles per day? c. Is it reasonable to think that this pattern will continue indefinitely? Explain.

Let's analyze Olivia's training plan step by step. ### **a. Equation for the \( n \)th Term of the Sequence** Olivia starts by running **3 miles per day in the first week**. Each subsequent week, she **increases her daily distance by half a mile**. This pattern forms an **arithmetic sequence** where: - **First Term (\( a_1 \))**: 3 miles - **Common Difference (\( d \))**: 0.5 miles The general formula for the \( n \)th term of an arithmetic sequence is: \[ a(n) = a_1 + (n - 1) \cdot d \] Plugging in the values: \[ a(n) = 3 + (n - 1) \cdot 0.5 \] **Simplified Form:** \[ a(n) = 0.5n + 2.5 \] So, the equation representing the \( n \)th term is: \[ a(n) = 3 + 0.5(n - 1) \] ### **b. Determining During Which Week She Will Run 10 Miles Per Day** We need to find the week number \( n \) when Olivia runs 10 miles per day. Using the equation from part **a**: \[ a(n) = 0.5n + 2.5 = 10 \] **Solving for \( n \):** \[ 0.5n + 2.5 = 10 \\ 0.5n = 10 - 2.5 \\ 0.5n = 7.5 \\ n = \frac{7.5}{0.5} \\ n = 15 \] **Answer:** Olivia will run **10 miles per day during the 15th week**. ### **c. Reasonableness of Continuing the Pattern Indefinitely** While the arithmetic sequence provides a clear pattern for increasing mileage, **continuing this pattern indefinitely is not reasonable** for several reasons: 1. **Physical Limitations:** The human body requires time to adapt to increased mileage. Continuous, unrestricted increases can lead to overuse injuries, fatigue, and burnout. 2. **Training Limits:** Marathon training typically involves progressive overload, but not at a constant linear rate. Training plans usually incorporate rest weeks and gradual ramp-ups to prevent injury and ensure optimal performance. 3. **Practical Constraints:** Running 0.5 additional miles each week would eventually lead to impractically high daily distances, making it unsustainable in the long term. **Conclusion:** It's essential to balance training intensity and rest. Olivia should consider incorporating rest periods and more sustainable mileage increases to prevent injury and ensure effective marathon preparation.