Let's analyze the sequence: $-5, 15, 45, -135$.

Recursive Equation

To find the recursive formula, we first observe the pattern in the sequence:

• Step 1: Find the common ratio between consecutive terms.

  $\frac{15}{-5} = -3, \quad \frac{45}{15} = 3, \quad \frac{-135}{45} = -3$

• Step 2: Notice that the sign alternates, indicating a multiplication by $-3$ for every two terms.

Thus, the recursive equation can be written as:

$a_{n} = -3 \cdot a_{n-1}$

where $a_1 = -5$.

Explicit Equation

For the explicit formula, we consider the formula for a geometric sequence:

$a_n = a_1 \cdot r^{n-1}$

However, in this case, the sign alternates. This suggests a formula of the form:

$a_n = -5 \times (-3)^{n-1}$

Final Equations:

• Recursive equation: $a_n = -3 \times a_{n-1}$, where $a_1 = -5$.
• Explicit equation: $a_n = -5 \times (-3)^{n-1}$.

Would you like more details or have any questions about this?

Related Questions:

1. How does the sign change affect the sequence in recursive equations?
2. What is the general form for an explicit formula in geometric sequences?
3. Can we generate the next two terms in this sequence?
4. How can you determine if a sequence is arithmetic or geometric?
5. What happens if the initial term $a_1$ is positive instead?

Tip:

Always check the ratio or difference between terms in a sequence to identify if it is geometric or arithmetic.