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Does the sequence below have a common difference or a common ratio?

The sequence provided is:

\[
2, 10, 50, 250, 1250, \ldots
\]

To determine whether it has a common difference (arithmetic sequence) or a common ratio (geometric sequence), let's analyze the terms.

1. **Check for common difference (arithmetic sequence):**

The difference between consecutive terms:

\[
10 - 2 = 8
\]
\[
50 - 10 = 40
\]
\[
250 - 50 = 200
\]
\[
1250 - 250 = 1000
\]

Since the differences are not the same, the sequence does **not** have a common difference. Thus, it is not an arithmetic sequence.

2. **Check for common ratio (geometric sequence):**

Now, calculate the ratio between consecutive terms:

\[
\frac{10}{2} = 5
\]
\[
\frac{50}{10} = 5
\]
\[
\frac{250}{50} = 5
\]
\[
\frac{1250}{250} = 5
\]

Since the ratio is consistently 5, the sequence has a **common ratio** of 5, making it a **geometric sequence**.

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:
1. How would you find the nth term of a geometric sequence?
2. Can you calculate the 7th term of this sequence?
3. How does an arithmetic sequence differ from a geometric sequence in terms of general formula?
4. How can you determine if a sequence is geometric just by looking at the terms?
5. What is the sum of the first 5 terms of this sequence?

**Tip:** In geometric sequences, you can quickly identify the type by checking the ratio between consecutive terms instead of focusing on differences.

What is the starting number in this sequence? 2, 10, 50, 250, 1250, ...

Learn how to identify the starting number and common ratio in the geometric sequence 2, 10, 50, 250, 1250. This problem explores geometric sequences and key formulas like a_n = a_1 * r^(n-1). Suitable for students in Grades 8-10.

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