Certainly! Let's break down the solution step-by-step for the given problem where the population grows according to an exponential growth model, with [tex]\( P_0 = 60 \)[/tex] and [tex]\( P_1 = 102 \)[/tex].
### Step 1: Determine the Multiplication Factor
To find the multiplication factor used in the recursive formula, we use the initial values [tex]\( P_0 \)[/tex] and [tex]\( P_1 \)[/tex]:
[tex]\[ P_0 = 60 \][/tex]
[tex]\[ P_1 = 102 \][/tex]
The multiplication factor is calculated as the ratio of [tex]\( P_1 \)[/tex] to [tex]\( P_0 \)[/tex]:
[tex]\[ \text{Multiplication Factor} = \frac{P_1}{P_0} = \frac{102}{60} = 1.7 \][/tex]
### Step 2: Write the Recursive Formula
Using the multiplication factor, the recursive formula can be written as:
[tex]\[ P_n = 1.7 \times P_{n-1} \][/tex]
### Step 3: Determine the Explicit Formula
The explicit formula for [tex]\( P_n \)[/tex] in an exponential growth model can be written using the initial population [tex]\( P_0 \)[/tex] and the multiplication factor. The formula is:
[tex]\[ P_n = P_0 \times (\text{Multiplication Factor})^n \][/tex]
Given [tex]\( P_0 = 60 \)[/tex] and the multiplication factor is [tex]\( 1.7 \)[/tex], the explicit formula becomes:
[tex]\[ P_n = 60 \times (1.7)^n \][/tex]
### Final Answer
Recursive Formula:
[tex]\[ P_n = 1.7 \times P_{n-1} \][/tex]
Explicit Formula:
[tex]\[ P_n = 60 \times (1.7)^n \][/tex]
