Question
Find the recursive formula for the geometric sequence −1/9
, −1/3
, −1
, −3
, −9
, . . .(1 point)
Responses
a1=−1/9
, an=−1/3an−1
a subscript 1 baseline equals negative Start Fraction 1 over 9 End Fraction , a subscript n baseline equals negative Start Fraction 1 over 3 End Fraction a subscript n minus 1 baseline
a1=−1/9
, an=3an−1
a subscript 1 baseline equals negative Start Fraction 1 over 9 End Fraction , a subscript n baseline equals 3 a subscript n minus 1 baseline
a1=−1/9
, an=−3an−1
a subscript 1 baseline equals negative Start Fraction 1 over 9 End Fraction , a subscript n baseline equals negative 3 a subscript n minus 1 baseline
a1=−1/9
, an=1/3an−1
, −1/3
, −1
, −3
, −9
, . . .(1 point)
Responses
a1=−1/9
, an=−1/3an−1
a subscript 1 baseline equals negative Start Fraction 1 over 9 End Fraction , a subscript n baseline equals negative Start Fraction 1 over 3 End Fraction a subscript n minus 1 baseline
a1=−1/9
, an=3an−1
a subscript 1 baseline equals negative Start Fraction 1 over 9 End Fraction , a subscript n baseline equals 3 a subscript n minus 1 baseline
a1=−1/9
, an=−3an−1
a subscript 1 baseline equals negative Start Fraction 1 over 9 End Fraction , a subscript n baseline equals negative 3 a subscript n minus 1 baseline
a1=−1/9
, an=1/3an−1
Answers
GPT-4o mini
To determine the recursive formula for the given geometric sequence:
\(-\frac{1}{9}, -\frac{1}{3}, -1, -3, -9, \ldots\)
First, we identify the first term \(a_1\).
- The first term \(a_1 = -\frac{1}{9}\).
Next, we need to derive the common ratio \(r\) by dividing any term by its preceding term. We can use the first two terms to calculate the common ratio:
\[
r = \frac{a_2}{a_1} = \frac{-\frac{1}{3}}{-\frac{1}{9}} = \frac{-\frac{1}{3} \cdot -9}{1} = 3
\]
We can confirm that the ratio remains consistent with subsequent terms:
\[
r = \frac{a_3}{a_2} = \frac{-1}{-\frac{1}{3}} = 3
\]
\[
r = \frac{a_4}{a_3} = \frac{-3}{-1} = 3
\]
\[
r = \frac{a_5}{a_4} = \frac{-9}{-3} = 3
\]
Now we see that the common ratio \(r = 3\).
The recursive formula for a geometric sequence is given by:
\[
a_n = r \cdot a_{n-1}
\]
Thus, we can express the recursive formula for this sequence as:
\[
a_1 = -\frac{1}{9}
\]
\[
a_n = 3 \cdot a_{n-1} \quad \text{for } n \geq 2
\]
This corresponds to the response:
**a1=−1/9, an=3an−1**
So the correct option is:
**a1=−1/9, an=3an−1**.
\(-\frac{1}{9}, -\frac{1}{3}, -1, -3, -9, \ldots\)
First, we identify the first term \(a_1\).
- The first term \(a_1 = -\frac{1}{9}\).
Next, we need to derive the common ratio \(r\) by dividing any term by its preceding term. We can use the first two terms to calculate the common ratio:
\[
r = \frac{a_2}{a_1} = \frac{-\frac{1}{3}}{-\frac{1}{9}} = \frac{-\frac{1}{3} \cdot -9}{1} = 3
\]
We can confirm that the ratio remains consistent with subsequent terms:
\[
r = \frac{a_3}{a_2} = \frac{-1}{-\frac{1}{3}} = 3
\]
\[
r = \frac{a_4}{a_3} = \frac{-3}{-1} = 3
\]
\[
r = \frac{a_5}{a_4} = \frac{-9}{-3} = 3
\]
Now we see that the common ratio \(r = 3\).
The recursive formula for a geometric sequence is given by:
\[
a_n = r \cdot a_{n-1}
\]
Thus, we can express the recursive formula for this sequence as:
\[
a_1 = -\frac{1}{9}
\]
\[
a_n = 3 \cdot a_{n-1} \quad \text{for } n \geq 2
\]
This corresponds to the response:
**a1=−1/9, an=3an−1**
So the correct option is:
**a1=−1/9, an=3an−1**.