Question

Read the story and respond to the requests below. Use the Format Editor x−−√ to type math or use the Image icon to upload a photo of your work.

The first person generates 5 emails to raise money for a fundraiser. If each of the five recipients sends the email to five more people, 25 emails are generated. If each of the twenty-five recipients sends five emails, 125 emails are generated.

Write the sequence represented by the story.
Is the sequence arithmetic or geometric?
Does it have a common difference or common ratio?
What is the common difference or common ratio?
What are the next three terms?

Answer 1

To analyze the story and determine the sequence, we start with the information given:

1. The first person sends **5** emails.
2. Each recipient sends **5** more emails, leading to subsequent generations of emails.

Here's the sequence indicated by the story:

- **First term (n=0)**: \(5\)
- **Second term (n=1)**: \(5 \times 5 = 25\)
- **Third term (n=2)**: \(25 \times 5 = 125\)

Continuing this pattern, we realize that each term is generated by multiplying the previous term by 5.

### Sequence Representation
The sequence is:
\[ 5, 25, 125, \ldots \]

### Type of Sequence
- This sequence is **geometric** because each term is multiplied by a constant factor to get the next term.

### Common Ratio
- The common ratio is \(5\) (each term is obtained by multiplying the previous term by 5).

### Next Three Terms
Using the common ratio of \(5\):
- Fourth term: \(125 \times 5 = 625\)
- Fifth term: \(625 \times 5 = 3125\)
- Sixth term: \(3125 \times 5 = 15625\)

### Summary
1. **Sequence**: \(5, 25, 125, 625, 3125, 15625\)
2. **Type**: Geometric
3. **Common Ratio**: \(5\)
4. **Next three terms**: \(625, 3125, 15625\)