An athlete trains by running 600 metres on the first day, 900 metres on the second, 1200 metres on the third and so forth. (a) How far does he run on the 15 th day? (b) What is the total distance that he will run in 15 days? (c) How long will it be before he can run a marathon of 42 km ?

Let's analyze the athlete's training regimen step by step. ### Given: - **First day distance (a₁):** 600 meters - **Daily increase (d):** 300 meters This forms an **arithmetic sequence** where each day's running distance increases by a constant amount. --- ### **a) Distance on the 15th Day** The formula for the **n-th term** of an arithmetic sequence is: \[ a_n = a_1 + (n - 1) \times d \] Substituting the values for the 15th day: \[ \begin{align*} a_{15} &= 600 + (15 - 1) \times 300 \\ &= 600 + 14 \times 300 \\ &= 600 + 4200 \\ &= 4800 \text{ meters} \end{align*} \] **Answer:** On the 15th day, he runs **4,800 meters**. --- ### **b) Total Distance in 15 Days** The formula for the **sum of the first n terms (Sₙ)** of an arithmetic sequence is: \[ S_n = \frac{n}{2} \times (a_1 + a_n) \] Using the 15th day's distance from part (a): \[ \begin{align*} S_{15} &= \frac{15}{2} \times (600 + 4800) \\ &= \frac{15}{2} \times 5400 \\ &= 15 \times 2700 \\ &= 40,500 \text{ meters} \end{align*} \] **Answer:** Over 15 days, he runs a total of **40,500 meters** (or **40.5 kilometers**). --- ### **c) Time to Run a Marathon (42 km)** A marathon is **42 kilometers**, which is **42,000 meters**. We need to find the day **n** when he first runs **≥ 42,000 meters**. Using the n-th term formula: \[ \begin{align*} a_n &\geq 42,000 \\ 600 + (n - 1) \times 300 &\geq 42,000 \\ (n - 1) \times 300 &\geq 42,000 - 600 \\ (n - 1) \times 300 &\geq 41,400 \\ n - 1 &\geq \frac{41,400}{300} \\ n - 1 &\geq 138 \\ n &\geq 139 \end{align*} \] So, on the **139th day**, he runs exactly **42,000 meters**. **Answer:** He will be able to run a marathon of **42 km on the 139th day** of training. --- ### **Summary of Answers:** - **a)** 4,800 meters - **b)** 40,500 meters - **c)** 139 days