Find the recursive formula for the geometric sequence −1/9, −1/3, −1, −3, −9, ...

Question

(1 point)
Responses

a1=−1/9, an=1/3an−1
a1=−1/9, an=1/3an−1

a1=−1/9, an=−3an−1
a1=−1/9, an=−3an−1

a1=−1/9, an=3an−1
a1=−1/9, an=3an−1

a1=−1/9, an=−1/3an−1

Answer 1

To identify the recursive formula for the given geometric sequence:

Given the sequence: \(-\frac{1}{9}, -\frac{1}{3}, -1, -3, -9, \ldots\)

1. Identify the first term:

   • The first term \(a_1\) is \(-\frac{1}{9}\).

2. Find the common ratio:

   • To find the common ratio \(r\), we can divide the second term by the first term: \[ r = \frac{-\frac{1}{3}}{-\frac{1}{9}} = \frac{-1/3 \times -9/1}{1} = 3 \]

3. Write the recursive formula:

   • The recursive formula of a geometric sequence can be written as: \[ a_n = r \cdot a_{n-1} \]
   • Here, \(r = 3\). Thus, the recursive formula is: \[ a_n = 3a_{n-1} \]

Putting it all together, the recursive formula is:

\[ a_1 = -\frac{1}{9}, \quad a_n = 3a_{n-1} \]

The correct response is:

\[ a_1 = -\frac{1}{9}, \quad a_n = 3a_{n-1} \]