Given the sequence, 5, 12.5, 31.25, 78.125, … Which of the following sequences has the same rule?

(1 Point)

3, 10.5, 18.25, 25.5

-1, -2.5, -6.25, -15.625

8, 10.5, 13.25, 15.75

16, 6.4, 2.56, 1.024

The given sequence is: 

\[ 5, 12.5, 31.25, 78.125, \dots \]

We can see that each term is obtained by multiplying the previous term by a constant factor of \( 2.5 \):

- \( 5 \times 2.5 = 12.5 \)
- \( 12.5 \times 2.5 = 31.25 \)
- \( 31.25 \times 2.5 = 78.125 \)

This is a geometric sequence where the common ratio is \( 2.5 \).

Now, let's analyze the options:

1. **3, 10.5, 18.25, 25.5**: This sequence increases, but the differences between terms are not constant multiples (it looks more like an arithmetic progression), so it does not follow a geometric rule.

2. **-1, -2.5, -6.25, -15.625**: Each term is obtained by multiplying the previous term by \( 2.5 \), which matches the rule of the original sequence.

3. **8, 10.5, 13.25, 15.75**: The differences between terms are increasing, but not by a constant factor. This appears to be an arithmetic sequence, not geometric.

4. **16, 6.4, 2.56, 1.024**: Each term is multiplied by \( 0.4 \), which is a geometric sequence, but the ratio is not \( 2.5 \).

Thus, the sequence that follows the same rule as the given sequence is:

**-1, -2.5, -6.25, -15.625**

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Would you like any further details on this? Here are 5 questions you might explore based on this topic:

1. How do you identify if a sequence is arithmetic or geometric?
2. Can the common ratio of a geometric sequence be negative?
3. How would you find the 6th term in the original sequence?
4. What is the general formula for the nth term of a geometric sequence?
5. How do you determine the sum of the first n terms of a geometric sequence?

**Tip:** For geometric sequences, the ratio between consecutive terms is constant, which is key to recognizing the pattern.

Discover how to identify geometric sequences by analyzing the sequence 5, 12.5, 31.25, 78.125 and recognizing the common ratio of 2.5. This example includes multiple-choice options with detailed explanations, focusing on understanding geometric sequence patterns. Suitable for students in Grades 8-10.

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