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Find the recursive formula for the geometric sequence −19

−
1
9
, −13
−
1
3
, −1
−
1
, −3
−
3
, −9
−
9
, . . .(1 point)
Responses

a1=−19
𝑎
1
=
−
1
9
, an=13an−1
𝑎
𝑛
=
1
3
𝑎
𝑛
−
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
𝑎
1
=
−
1
9
, an=3an−1
𝑎
𝑛
=
3
𝑎
𝑛
−
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
𝑎
1
=
−
1
9
, an=−3an−1
𝑎
𝑛
=
−
3
𝑎
𝑛
−
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
𝑎
1
=
−
1
9
, an=−13an−1

1 answer

To find the recursive formula for the geometric sequence given:

−19, −13/3, −1/9, −3/3, −9/9, ...

We first need to identify the first term and the common ratio.

  1. The first term \( a_1 = -19 \).

  2. To find the common ratio \( r \), we can divide any term by the preceding term. Let's calculate the ratio using the first two terms:

\[ r = \frac{a_2}{a_1} = \frac{-13/3}{-19} = \frac{13}{3 \cdot 19} = \frac{13}{57} \]

This suggests that our common ratio is \( r = \frac{13}{57} \). However, the values given in the sequence do not appear to consistently provide a clear geometric progression; rather, they fluctuate in sign and magnitude without a straightforward solution.

Since the values appear to be alternating and may indicate a different pattern, we should recalculate if the ratio is consistent throughout. Calculating from further terms does show fluctuations, suggesting possible mistakes rather than straightforward terms.

Ultimately, upon examining the available response options, the most suitable recursive definition appears to include the first term and a common ratio.

Thus, if we stick to the given multiple-choice options, the correct formula in the context of your problem seems to reflect this iterative relationship, most closely aligning with slight adjustments considering the context you provided.

Among the multiple-choice responses offered, the best representation technically appears to be:

  • Given \( a_1 = -\frac{1}{9}, \text{and} , a_n = -3a_{n-1} \)

Final Recursive Formula:

  • \( a_1 = -19 \), and \( a_n = -3 a_{n-1} \)

This reflects the first condition, establishing \( a_1 \) and the recursive relationship as indicated above.