Question: A population grows according to an exponential growth model, with \( P_0 = 50 \) and \( P_1 = 95…

A population grows according to an exponential growth model, with \( P_0 = 50 \) and \( P_1 = 95 \).

Complete the recursive formula:

\[ P_n = \boxed{\phantom{0}} \times P_{n-1} \]

Write an explicit formula for \( P_n \)

\[ P_n = \boxed{\phantom{0}} \]

Solution

Step 1: Determine the common ratio. \[ P_1 = \text{Multiplier} \times P_0 & \] \[ 95 = \text{Multiplier} \times 50 & \] Step 2: Solve for the multiplier. \[ \text{Multiplier} = \frac{95}{50} = 1.9 & \] Step 3: Write the recursive formula. \[ P_n = 1.9 \times P_{n-1} \] Step 4: Derive the explicit formula. \[ P_n = 50 \times (1.9)^n \]