Which recursive formula can be used to define this sequence for n>1?
2, 6, 18, 54, 162, 486, ...


The given sequence is: 
\[ 2, 6, 18, 54, 162, 486, \dots \]

To find a recursive formula, observe the pattern:

- \( \frac{6}{2} = 3 \)
- \( \frac{18}{6} = 3 \)
- \( \frac{54}{18} = 3 \)
- \( \frac{162}{54} = 3 \)
- \( \frac{486}{162} = 3 \)

This shows that each term is obtained by multiplying the previous term by 3. Therefore, the recursive formula can be expressed as:

\[
a_n = 3 \cdot a_{n-1} \quad \text{for} \quad n > 1
\]

with the initial condition:

\[
a_1 = 2
\]

Thus, the recursive formula for the sequence is:

\[
a_n = 3 \cdot a_{n-1}, \quad \text{where} \quad a_1 = 2
\]

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to expand on this concept:

1. How can we generalize a recursive formula for any geometric sequence?
2. What is the explicit (non-recursive) formula for this sequence?
3. How do you determine whether a sequence is arithmetic or geometric?
4. Can you provide an example of a recursive sequence that decreases over time?
5. What is the sum of the first \( n \) terms in a geometric sequence?

**Tip:** In geometric sequences, the ratio between consecutive terms (called the common ratio) is constant, which makes finding recursive formulas straightforward.

Which recursive formula can be used to define this sequence for n>1? 2, 6, 18, 54, 162, 486, ...

Explore how to find a recursive formula for the geometric sequence 2, 6, 18, 54, 162, 486, ... Learn the key concepts behind geometric sequences and how to define them using recursion. Suitable for students in Grades 9-10.

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