Identify Recursive Formula for Geometric Sequence: aₙ = 2

Math Problem Statement

Answer

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

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.

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).

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

$a_n = r \cdot a_{n-1}$

where:

Thus, the recursive version of this sequence would be:

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.

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Sequences
Geometric Sequences
Recursion

aₙ = r · aₙ₋₁ (recursive formula for geometric sequences)
aₙ = a₁ · r⁽ⁿ⁻¹⁾ (explicit formula for geometric sequences)

Geometric Sequence Common Ratio

Grades 9-12

Identify the Recursive Formula for a Geometric Sequence: a_n = 2 * 5^(n-1)

Geometric Sequence: Deriving Explicit Formulas from Recurrence Relations

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Recursive Formula for the Geometric Sequence: 2, 6, 18, 54