Answer :
To solve for a population that grows according to an exponential growth model, given the initial conditions [tex]\( P_0 = 20 \)[/tex] and [tex]\( P_1 = 28 \)[/tex], we first need to determine the growth factor and then use it to complete both the recursive and explicit formulas.
### Step-by-Step Solution:
1. Determine the Growth Factor:
To find the growth factor, we compare the population at time [tex]\( n = 1 \)[/tex] to the population at time [tex]\( n = 0 \)[/tex]:
[tex]\[ \text{Growth Factor} = \frac{P_1}{P_0} = \frac{28}{20} = 1.4 \][/tex]
2. Complete the Recursive Formula:
The recursive formula is of the form:
[tex]\[ P_n = \text{Growth Factor} \times P_{n-1} \][/tex]
Substituting the growth factor we found:
[tex]\[ P_n = 1.4 \times P_{n-1} \][/tex]
3. Write the Explicit Formula:
The explicit formula for an exponential growth model can be written as:
[tex]\[ P_n = P_0 \times (\text{Growth Factor})^n \][/tex]
Substituting the known values:
[tex]\[ P_n = 20 \times (1.4)^n \][/tex]
To summarize:
1. The recursive formula is:
[tex]\[ P_n = 1.4 \times P_{n-1} \][/tex]
2. The explicit formula is:
[tex]\[ P_n = 20 \times (1.4)^n \][/tex]
### Step-by-Step Solution:
1. Determine the Growth Factor:
To find the growth factor, we compare the population at time [tex]\( n = 1 \)[/tex] to the population at time [tex]\( n = 0 \)[/tex]:
[tex]\[ \text{Growth Factor} = \frac{P_1}{P_0} = \frac{28}{20} = 1.4 \][/tex]
2. Complete the Recursive Formula:
The recursive formula is of the form:
[tex]\[ P_n = \text{Growth Factor} \times P_{n-1} \][/tex]
Substituting the growth factor we found:
[tex]\[ P_n = 1.4 \times P_{n-1} \][/tex]
3. Write the Explicit Formula:
The explicit formula for an exponential growth model can be written as:
[tex]\[ P_n = P_0 \times (\text{Growth Factor})^n \][/tex]
Substituting the known values:
[tex]\[ P_n = 20 \times (1.4)^n \][/tex]
To summarize:
1. The recursive formula is:
[tex]\[ P_n = 1.4 \times P_{n-1} \][/tex]
2. The explicit formula is:
[tex]\[ P_n = 20 \times (1.4)^n \][/tex]
