Is the pattern 2, -6, 18 -54 explicit or recursive

The sequence **2, -6, 18, -54** can be described in both **explicit** and **recursive** forms. Here's how:

### 1. **Recursive Formula**:
A recursive formula defines each term in relation to the previous one. In this sequence, you can see that each term is multiplied by **-3** to get the next term:
- \(-6 = 2 \times -3\)
- \(18 = -6 \times -3\)
- \(-54 = 18 \times -3\)

So, the recursive relation is:
\[
a_n = a_{n-1} \times (-3)
\]
with the first term \(a_1 = 2\).

### 2. **Explicit Formula**:
An explicit formula expresses the \(n\)-th term directly in terms of \(n\), without reference to previous terms. The sequence follows a geometric progression with the first term \(a_1 = 2\) and the common ratio \(r = -3\). The explicit formula for the \(n\)-th term of a geometric sequence is:
\[
a_n = a_1 \times r^{n-1}
\]
Substituting the known values:
\[
a_n = 2 \times (-3)^{n-1}
\]

Both forms are valid for describing the pattern.

Would you like more details on either approach or how to apply these formulas?

Here are 5 questions related to the sequence:
1. What is the 6th term of the sequence using the recursive formula?
2. How do you derive the 10th term using the explicit formula?
3. Can you find a different geometric sequence with a similar pattern?
4. What would change if the common ratio was positive instead of negative?
5. How do recursive and explicit formulas differ in practical applications?

**Tip**: For quick calculations, explicit formulas are often more efficient, but recursive ones help understand how each term is built step-by-step.

Is the pattern 2, -6, 18, -54 explicit or recursive?

Explore whether the sequence 2, -6, 18, -54 can be described using explicit or recursive formulas. This problem delves into key algebraic concepts like geometric progressions and recursive formulas, making it ideal for students in Grades 9-11.

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