Question

A geometric sequence starts with 8. Each term is the previous term multiplied by −38 . What is the recursive formula? What is the explicit formula?(1 point) Recursive: a1= ; an=a(n−1)⋅ (− $$ ) Explicit: an= ⋅ (− $$ )(n−1)

Answer 1

Let's denote the first term of the geometric sequence as \( a_1 \).

Given that the first term is 8 and that each subsequent term is multiplied by \(-38\), we can find the recursive and explicit formulas.

### Recursive Formula

1. The first term \( a_1 \) is given as:
\[
a_1 = 8
\]
2. For the recursive definition, we can express the subsequent terms:
\[
a_n = a_{n-1} \cdot (-38) \quad \text{for } n \geq 2
\]

So, the complete recursive formula is:
\[
\text{Recursive: } a_1 = 8; \quad a_n = a_{n-1} \cdot (-38) \text{ for } n \geq 2
\]

### Explicit Formula

The explicit formula for the \( n \)-th term of a geometric sequence can be expressed using the formula:
\[
a_n = a_1 \cdot r^{n-1}
\]
where \( a_1 \) is the first term and \( r \) is the common ratio (which is \(-38\) in this case).

1. Substituting the values:
\[
a_n = 8 \cdot (-38)^{n-1}
\]

So, the complete explicit formula is:
\[
\text{Explicit: } a_n = 8 \cdot (-38)^{n-1}
\]

### Summary

- **Recursive:** \( a_1 = 8; \; a_n = a_{n-1} \cdot (-38) \text{ for } n \geq 2 \)
- **Explicit:** \( a_n = 8 \cdot (-38)^{n-1} \)