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Question

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: \(625, 3125, 15625\)