Question
Find the recursive formula for the geometric sequence −19, −13 , −1 , −3 , −9 , . . .(1 point)
Responses
a1=−19
, an=−13an−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=−19
, 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=−19
, 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=−19
, an=13an−1
Responses
a1=−19
, an=−13an−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=−19
, 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=−19
, 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=−19
, an=13an−1
Answers
GPT-4o mini
To find the recursive formula for the given geometric sequence, we first need to identify the first term \(a_1\) and the common ratio \(r\).
The first term \(a_1\) in the sequence is:
\[ a_1 = -19 \]
Next, we will determine the common ratio by dividing the second term by the first term:
\[ r = \frac{a_2}{a_1} = \frac{-13}{-19} = \frac{13}{19} \]
We can also check the ratio between other consecutive terms:
\[ r = \frac{a_3}{a_2} = \frac{-1}{-13} = \frac{1}{13} \]
\[ r = \frac{a_4}{a_3} = \frac{-3}{-1} = 3 \]
\[ r = \frac{a_5}{a_4} = \frac{-9}{-3} = 3 \]
The values for \(r\) are not consistent, suggesting that this might not be a purely geometrical sequence.
To express it in a recursive formula, we recognize that the possible answers all use the structure \(a_n = k \cdot a_{n-1}\) where \(k\) is a multiplier.
Since there are inconsistencies in the results from the ratios, let's iterate through the provided possible answers to check for a match.
1. **\(a_1 = -19, a_n = -13 a_{n-1}\)** (Not valid since it doesn't match the sequence.)
2. **\(a_1 = -19, a_n = -3 a_{n-1}\)** (Not valid since it doesn't match the sequence.)
3. **\(a_1 = -19, a_n = 3 a_{n-1}\)** (Not valid since it doesn't match the sequence.)
4. **\(a_1 = -19, a_n = 13 a_{n-1}\)** (Not valid since it doesn't match the sequence.)
None of the provided answers seem to correctly describe the recursion based on the observed sequence.
Thus, while the first term \(a_1 = -19\) is correct, there seems to be a misunderstanding or misrepresentation in the options given for \(a_n\). The recursive definition does not fit a simple ratio.
Given the terms provided, none of the given options accurately represent the subsequent terms in a consistent format.
The first term \(a_1\) in the sequence is:
\[ a_1 = -19 \]
Next, we will determine the common ratio by dividing the second term by the first term:
\[ r = \frac{a_2}{a_1} = \frac{-13}{-19} = \frac{13}{19} \]
We can also check the ratio between other consecutive terms:
\[ r = \frac{a_3}{a_2} = \frac{-1}{-13} = \frac{1}{13} \]
\[ r = \frac{a_4}{a_3} = \frac{-3}{-1} = 3 \]
\[ r = \frac{a_5}{a_4} = \frac{-9}{-3} = 3 \]
The values for \(r\) are not consistent, suggesting that this might not be a purely geometrical sequence.
To express it in a recursive formula, we recognize that the possible answers all use the structure \(a_n = k \cdot a_{n-1}\) where \(k\) is a multiplier.
Since there are inconsistencies in the results from the ratios, let's iterate through the provided possible answers to check for a match.
1. **\(a_1 = -19, a_n = -13 a_{n-1}\)** (Not valid since it doesn't match the sequence.)
2. **\(a_1 = -19, a_n = -3 a_{n-1}\)** (Not valid since it doesn't match the sequence.)
3. **\(a_1 = -19, a_n = 3 a_{n-1}\)** (Not valid since it doesn't match the sequence.)
4. **\(a_1 = -19, a_n = 13 a_{n-1}\)** (Not valid since it doesn't match the sequence.)
None of the provided answers seem to correctly describe the recursion based on the observed sequence.
Thus, while the first term \(a_1 = -19\) is correct, there seems to be a misunderstanding or misrepresentation in the options given for \(a_n\). The recursive definition does not fit a simple ratio.
Given the terms provided, none of the given options accurately represent the subsequent terms in a consistent format.