1. **Problem statement:** Jason observes that the amount of money donated each hour increases by 15 compared to the previous hour, starting with 40 in the first hour. We need to: a) Write a function $D(n)$ for the amount donated in the $n^{th}$ hour. b) Find the total amount donated in the first 5 hours. c) Find the difference between the amount donated in the 5th hour and the 2nd hour. 2. **Writing the function $D(n)$:** Since the donation increases by a constant amount each hour, this is an arithmetic sequence. The formula for the $n^{th}$ term of an arithmetic sequence is: $$D(n) = D(1) + (n-1) \times d$$ where $D(1) = 40$ and the common difference $d = 15$. So, $$D(n) = 40 + (n-1) \times 15$$ 3. **Calculating the total amount donated in the first 5 hours:** The total amount is the sum of the first 5 terms of the arithmetic sequence. The sum of the first $n$ terms is: $$S_n = \frac{n}{2} \times (D(1) + D(n))$$ First, find $D(5)$: $$D(5) = 40 + (5-1) \times 15 = 40 + 4 \times 15 = 40 + 60 = 100$$ Then, $$S_5 = \frac{5}{2} \times (40 + 100) = \frac{5}{2} \times 140 = \cancel{\frac{5}{2}} \times 140 = 5 \times 70 = 350$$ 4. **Calculating the difference between the 5th and 2nd hour donations:** Find $D(2)$: $$D(2) = 40 + (2-1) \times 15 = 40 + 15 = 55$$ Difference: $$D(5) - D(2) = 100 - 55 = 45$$ **Final answers:** - a) $D(n) = 40 + 15(n-1)$ - b) Total donated in first 5 hours is 350 - c) Difference between 5th and 2nd hour donations is 45