\( \begin{array}{ll} & \text { Consider a population that grows according to the recursive rule } P_{n}=P_{n-1}+50 \text {, with } \\ \text { initial population } P_{0}=60 \text {. }\end{array} \)

Certainly! Let's analyze the population growth according to the given recursive rule. ### Given: - **Recursive Formula:** \( P_n = P_{n-1} + 50 \) - **Initial Population:** \( P_0 = 60 \) ### Understanding the Recursive Formula: The recursive formula indicates that each subsequent population \( P_n \) increases by 50 individuals from the previous population \( P_{n-1} \). This is characteristic of an **arithmetic sequence**, where each term increases by a fixed amount. ### Deriving the Explicit Formula: While the recursive formula defines each term based on the previous one, an explicit formula allows us to calculate \( P_n \) directly without knowing \( P_{n-1} \). For an arithmetic sequence, the \( n \)-th term can be expressed as: \[ P_n = P_0 + d \cdot n \] where: - \( P_0 \) is the initial term, - \( d \) is the common difference (in this case, 50), - \( n \) is the number of periods (e.g., years, months). Applying the values: \[ P_n = 60 + 50n \] ### Examples: Let's compute the population for the first few periods. 1. **First Period (\( n = 1 \)):** \[ P_1 = 60 + 50 \times 1 = 110 \] 2. **Second Period (\( n = 2 \)):** \[ P_2 = 60 + 50 \times 2 = 160 \] 3. **Third Period (\( n = 3 \)):** \[ P_3 = 60 + 50 \times 3 = 210 \] 4. **...and so on.** ### General Insights: - **Linear Growth:** Since the population increases by a fixed amount each period, the growth is linear. - **Slope Interpretation:** In the explicit formula \( P_n = 60 + 50n \), the coefficient \( 50 \) represents the rate of increase per period. - **Time to Reach a Specific Population:** If you want to find out after how many periods the population reaches a certain size, you can set \( P_n \) to that size and solve for \( n \). ### Example Problem: **Question:** After how many periods will the population reach 310? **Solution:** Set \( P_n = 310 \): \[ 310 = 60 + 50n \] \[ 310 - 60 = 50n \] \[ 250 = 50n \] \[ n = 5 \] **Answer:** After **5 periods**, the population will reach 310. ### Summary: - **Recursive Formula:** Defines each term based on the previous one. - **Explicit Formula:** Allows direct computation of any term without referencing previous terms. - **Arithmetic Sequence Characteristics:** Fixed increase per period, leading to linear growth. If you have a specific question or need further elaboration on a particular aspect of this population model, feel free to ask!