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 Start Fraction 1 over 3 End Fraction a subscript n minus 1 baseline
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 negative 3 a subscript n minus 1 baseline
a1=−1/9
, an=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 Start Fraction 1 over 3 End Fraction a subscript n minus 1 baseline
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 negative 3 a subscript n minus 1 baseline
a1=−1/9
, an=3an−1
Answers
GPT-4o mini
To find the recursive formula for the given geometric sequence:
\(-\frac{1}{9}, -\frac{1}{3}, -1, -3, -9, \ldots\)
First, we identify the first term and the common ratio.
1. The first term \(a_1\) is \(-\frac{1}{9}\).
2. To find the common ratio \(r\), we can divide the second term by the first term:
\[
r = \frac{-\frac{1}{3}}{-\frac{1}{9}} = \frac{1/3}{1/9} = \frac{1}{3} \times \frac{9}{1} = 3.
\]
So, the common ratio is \(3\).
The general form of the recursive definition for a geometric sequence is:
- The first term is given: \(a_1 = -\frac{1}{9}\).
- Each subsequent term can be defined as: \(a_n = r \cdot a_{n-1}\), where \(r\) is the common ratio.
In this case, the recursive relation using the common ratio \(3\) is:
\[
a_n = 3 \cdot a_{n-1}.
\]
Putting this all together, the correct recursive formula is:
\[
\text{ } a_1 = -\frac{1}{9}, \; a_n = 3 \cdot a_{n-1}.
\]
None of the given options exactly match this statement. The closest option would involve the negative ratio (which is incorrect), so be cautious regarding signs. The correct recursive formula resolves to:
**a1 = -1/9, an = 3an−1** (not provided as an option).
\(-\frac{1}{9}, -\frac{1}{3}, -1, -3, -9, \ldots\)
First, we identify the first term and the common ratio.
1. The first term \(a_1\) is \(-\frac{1}{9}\).
2. To find the common ratio \(r\), we can divide the second term by the first term:
\[
r = \frac{-\frac{1}{3}}{-\frac{1}{9}} = \frac{1/3}{1/9} = \frac{1}{3} \times \frac{9}{1} = 3.
\]
So, the common ratio is \(3\).
The general form of the recursive definition for a geometric sequence is:
- The first term is given: \(a_1 = -\frac{1}{9}\).
- Each subsequent term can be defined as: \(a_n = r \cdot a_{n-1}\), where \(r\) is the common ratio.
In this case, the recursive relation using the common ratio \(3\) is:
\[
a_n = 3 \cdot a_{n-1}.
\]
Putting this all together, the correct recursive formula is:
\[
\text{ } a_1 = -\frac{1}{9}, \; a_n = 3 \cdot a_{n-1}.
\]
None of the given options exactly match this statement. The closest option would involve the negative ratio (which is incorrect), so be cautious regarding signs. The correct recursive formula resolves to:
**a1 = -1/9, an = 3an−1** (not provided as an option).