https://file.aiquickdraw.com/mathbot/file/1729632154885l5rs4cdw.jpg

Let's analyze the problem.

We are given the formula \( a_n = 2 \cdot 5^{(n-1)} \), and the question asks which recursive formula represents the same geometric sequence.

### Understanding the given formula:

The formula \( a_n = 2 \cdot 5^{(n-1)} \) describes a geometric sequence where:
- The first term \( a_1 = 2 \) (when \( n = 1 \)),
- Each subsequent term is multiplied by 5 (because the sequence has a common ratio of 5).

### Converting this into a recursive formula:

In a recursive formula for a geometric sequence, the general form is:

\[
a_n = r \cdot a_{n-1}
\]

where:
- \( r \) is the common ratio (which is 5 in this case),
- \( a_1 = 2 \) is the first term.

Thus, the recursive version of this sequence would be:

\[
\begin{cases}
a_1 = 2 \\
a_n = 5 \cdot a_{n-1}
\end{cases}
\]

### Now, looking at the options:

- **Option A:** 
  \[
  \begin{cases}
  a_1 = 2 \\
  a_n = 5 \cdot a_{n-1}
  \end{cases}
  \]
  This matches our recursive formula, so **Option A is correct**.

- **Option B:** 
  The common ratio is 10, which is incorrect.
  
- **Option C:** 
  The sequence starts with \( a_1 = 5 \) and adds 2, which is incorrect.

- **Option D:** 
  The common ratio is 2, which is incorrect.

### Final Answer: **Option A**

Do you have any questions or would you like more details?

Here are some related questions you might consider:
1. What is a geometric sequence, and how do you identify its common ratio?
2. How do you convert a general sequence formula into a recursive one?
3. What are the key differences between recursive and explicit formulas?
4. Can you explain why Option B is not correct despite the similarity in numbers?
5. What types of sequences can be described with recursive formulas?

**Tip:** Understanding the relationship between the explicit and recursive forms of a sequence is crucial for translating between them.

Which of the following recursive formulas represents the same geometric sequence as the formula aₙ = 2 · 5⁽ⁿ⁻¹⁾?

Understand how to determine the correct recursive formula for the geometric sequence represented by aₙ = 2 · 5⁽ⁿ⁻¹⁾. Learn key concepts in sequences, recursion, and the identification of common ratios. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation