In the movie “Pay It Forward,” a student, Trevor, comes up with an idea that he thought could change the world.  He decides to do a good deed for three people and then each of the three people would do a good deed for three more people and so on. He believed that before long there would be good things happening to billions of people. At stage 1 of the process, Trevor completes three good deeds. How does the number of good deeds grow from stage to stage?  How many good deeds would be completed at stage 5?  Describe a function that would model the Pay It Forward process at any stage.

The "Pay It Forward" process described by Trevor can be modeled as an exponential growth situation. Here's how it unfolds:

### Step-by-Step Explanation:

1. **Stage 1 (Trevor's actions):**
   - Trevor performs good deeds for **3 people**.

2. **Stage 2 (Each of those 3 people does good deeds for 3 more people):**
   - Each of the 3 people from Stage 1 performs good deeds for 3 more people. This results in \( 3 \times 3 = 9 \) good deeds.

3. **Stage 3 (Each of the 9 people does good deeds for 3 more people):**
   - Each of the 9 people from Stage 2 performs good deeds for 3 more people, resulting in \( 9 \times 3 = 27 \) good deeds.

4. **Stage 4 (Each of the 27 people does good deeds for 3 more people):**
   - Each of the 27 people from Stage 3 performs good deeds for 3 more people, resulting in \( 27 \times 3 = 81 \) good deeds.

5. **Stage 5 (Each of the 81 people does good deeds for 3 more people):**
   - Each of the 81 people from Stage 4 performs good deeds for 3 more people, resulting in \( 81 \times 3 = 243 \) good deeds.

### How the number of good deeds grows:
The number of good deeds follows an exponential pattern. The general pattern is that at each stage, the number of good deeds grows by a factor of 3.

- Stage 1: \( 3 \)
- Stage 2: \( 3^2 = 9 \)
- Stage 3: \( 3^3 = 27 \)
- Stage 4: \( 3^4 = 81 \)
- Stage 5: \( 3^5 = 243 \)

### General Function to Model the Process:
The function that models the number of good deeds at stage \( n \) is:
\[
f(n) = 3^n
\]
Where:
- \( f(n) \) is the number of good deeds completed at stage \( n \),
- \( n \) is the stage number.

### How many good deeds are completed at stage 5?
At stage 5, the total number of good deeds completed is:
\[
f(5) = 3^5 = 243
\]
So, 243 good deeds are completed at stage 5.

### Conclusion:
The number of good deeds grows exponentially, and the growth can be described by the function \( f(n) = 3^n \), where \( n \) is the stage. At stage 5, 243 good deeds are completed.

Would you like more details on the exponential growth process, or do you have any questions?

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Here are 5 related questions to consider:

1. What would the total number of good deeds be after **all** 5 stages, including stages 1 through 5?
2. How would the number of good deeds change if each person did good deeds for **4** people instead of 3?
3. At what stage would the number of good deeds exceed **1,000**?
4. How can exponential growth in this model be applied to real-world scenarios, like social media sharing or viral campaigns?
5. What are the limitations of this model if applied in real life?

**Tip:** Exponential growth can get very large, very quickly, even with small base numbers like 3!

In the movie 'Pay It Forward,' a student, Trevor, comes up with an idea that he thought could change the world. He decides to do a good deed for three people, and then each of the three people would do a good deed for three more people, and so on. He believed that before long there would be good things happening to billions of people. At stage 1 of the process, Trevor completes three good deeds. How does the number of good deeds grow from stage to stage? How many good deeds would be completed at stage 5? Describe a function that would model the Pay It Forward process at any stage.

Explore how the number of good deeds grows exponentially in the 'Pay It Forward' model. Learn how to model the process with the function f(n) = 3^n, where n is the stage, and see how 243 good deeds are completed by stage 5. Suitable for Grades 8-10, this problem covers key exponential growth concepts.

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