A population grows according to an exponential growth model. The initial population is  
P
0
=
16
 , and the growth rate is  
r
=
0.35
 .

Then:

P
1
  = 
21.6
Correct
P
2
  = 
29.2
Correct

Find an explicit formula for  
P
n
 . Your formula should involve  
n
 .

P
n
  = 
Incorrect  

Use your formula to find  
P
9
 

P
9
  = 
21.6
Incorrect

Give all answers accurate to at least one decimal place

To solve this problem, we start with the formula for exponential growth, which is:

\[ P_n = P_0 \cdot (1 + r)^n \]

where:
- \( P_n \) is the population at time \( n \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate, and
- \( n \) is the number of time intervals.

Given:
- \( P_0 = 16 \)
- \( r = 0.35 \)

The explicit formula for \( P_n \) is:

\[ P_n = 16 \cdot (1 + 0.35)^n \]

Now, let's calculate \( P_9 \):

\[ P_9 = 16 \cdot (1 + 0.35)^9 \]

\[ P_9 = 16 \cdot (1.35)^9 \]

Calculating the value:

\[ P_9 = 16 \cdot (1.35)^9 \]

\[ P_9 \approx 143.6 \]

So, the population at \( n = 9 \) is approximately 143.6.

**Explicit formula for \( P_n \)**: \( P_n = 16 \cdot (1.35)^n \)  
**Population at \( n = 9 \)**: \( P_9 \approx 143.6 \)

Would you like further details or have any other questions?

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### Related Questions

1. What would the population be at \( n = 5 \)?
2. How would the formula change if the initial population were different?
3. How does the growth rate \( r \) affect the population after a long period?
4. What is the population after 20 time intervals?
5. What happens to the population if the growth rate becomes negative?
6. How do you calculate the time it takes for the population to double?
7. What is the effect of compounding frequency on exponential growth?
8. Can exponential growth continue indefinitely in real-life scenarios?

**Tip:** When dealing with exponential growth, the base of the exponent (1 + r) indicates how quickly the quantity is increasing or decreasing.

Learn how to calculate population growth using the exponential growth model. Solve for the population at different intervals using the given initial population and growth rate. Example calculation provided for P_9 with an initial population of 16 and growth rate of 0.35.

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