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 Start Fraction 1 over 3 End Fraction a subscript n minus 1 baseline
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
, −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 Start Fraction 1 over 3 End Fraction a subscript n minus 1 baseline
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
Answers
GPT-4o mini
To determine the recursive formula for the geometric sequence given, we first need to identify the first term and the common ratio.
The sequence provided is:
-19, -13, -1, -3, -9, ...
1. **Identify the first term \( a_1 \)**:
\[
a_1 = -19
\]
2. **Calculate the common ratio \( r \)**:
The common ratio can be found by dividing any term by its previous term.
- For example, between the first (-19) and the second (-13):
\[
r = \frac{-13}{-19}
\]
We can perform the calculations:
\[
r = \frac{-13}{-19} = \frac{13}{19}
\]
Let's check the ratio between subsequent terms:
- From -13 to -1:
\[
r = \frac{-1}{-13} = \frac{1}{13}
\]
- From -1 to -3:
\[
r = \frac{-3}{-1} = 3
\]
- From -3 to -9:
\[
r = \frac{-9}{-3} = 3
\]
It appears that the ratios are not consistent, meaning the sequence is not geometric. Therefore, we must look at the terms to see if we can identify a different recursive relationship.
After closely checking the sequence:
- The first term is -19.
- The second is -13 (add 6).
- The third is -1 (add 12).
- The fourth is -3 (subtract 6).
- The fifth is -9 (subtract 6).
Based on the pattern:
- The third term (-1) is \( -13 + 12 \).
- The fourth term (-3) is indeed following a different operational step (not consistent).
This sequence does not fit the patterns given in geometric sequences.
Since it seems that the steps in changes are alternating between adding and subtracting, a more complex evaluation is necessary. However, all provided responses are of the geometric recursive form, which does not apply here.
Thus there is no suitable answer in the options since the sequence does not seem to adhere to a proper geometric format. The recursive relationship does not hold with any of the provided statements. If forced to select based on your choices, any of them would not apply correctly.
The sequence provided is:
-19, -13, -1, -3, -9, ...
1. **Identify the first term \( a_1 \)**:
\[
a_1 = -19
\]
2. **Calculate the common ratio \( r \)**:
The common ratio can be found by dividing any term by its previous term.
- For example, between the first (-19) and the second (-13):
\[
r = \frac{-13}{-19}
\]
We can perform the calculations:
\[
r = \frac{-13}{-19} = \frac{13}{19}
\]
Let's check the ratio between subsequent terms:
- From -13 to -1:
\[
r = \frac{-1}{-13} = \frac{1}{13}
\]
- From -1 to -3:
\[
r = \frac{-3}{-1} = 3
\]
- From -3 to -9:
\[
r = \frac{-9}{-3} = 3
\]
It appears that the ratios are not consistent, meaning the sequence is not geometric. Therefore, we must look at the terms to see if we can identify a different recursive relationship.
After closely checking the sequence:
- The first term is -19.
- The second is -13 (add 6).
- The third is -1 (add 12).
- The fourth is -3 (subtract 6).
- The fifth is -9 (subtract 6).
Based on the pattern:
- The third term (-1) is \( -13 + 12 \).
- The fourth term (-3) is indeed following a different operational step (not consistent).
This sequence does not fit the patterns given in geometric sequences.
Since it seems that the steps in changes are alternating between adding and subtracting, a more complex evaluation is necessary. However, all provided responses are of the geometric recursive form, which does not apply here.
Thus there is no suitable answer in the options since the sequence does not seem to adhere to a proper geometric format. The recursive relationship does not hold with any of the provided statements. If forced to select based on your choices, any of them would not apply correctly.