Question
Algebra 1 6-4 Quiz
7 of 7 7 of 7 Items
10:36 / 20:00
Question
The recursive formula for a geometric sequence is with the first term
. What is the explicit formula for the sequence?
Responses
A sub n is equal to 1 eighth times 4 raised to the n minus 1 power
A sub n is equal to 1 eighth times 2 raised to the n plus 1 power
A sub n is equal to 1 eighth times 2 raised to the n minus 1 power
A sub n is equal to 1 eighth times 4 raised to the n plus 1 power
Answers
Answered by
GPT-4o mini
To determine the explicit formula for the geometric sequence defined by its recursive formula, we need some key information: the first term and the common ratio. However, since the question does not specify a recursive formula or the values of the first term and common ratio, I will explain how to derive the explicit formula assuming:
- The first term of the sequence is \( a_1 = \frac{1}{8} \).
- The common ratio can be inferred from the answers given.
In general, for a geometric sequence, the explicit formula can be expressed as:
\[
A_n = a_1 \cdot r^{n-1}
\]
Where:
- \( A_n \) is the \( n \)-th term,
- \( a_1 \) is the first term of the sequence,
- \( r \) is the common ratio, and
- \( n \) is the term number.
From the response options provided, we need to analyze the forms:
1. \( A_n = \frac{1}{8} \cdot 4^{n-1} \)
2. \( A_n = \frac{1}{8} \cdot 2^{n+1} \)
3. \( A_n = \frac{1}{8} \cdot 2^{n-1} \)
4. \( A_n = \frac{1}{8} \cdot 4^{n+1} \)
To resolve which could be the explicit formula, we look at the bases involved. Notice that \( 4 \) can be expressed as \( 2^2 \). Thus, \( 4^{n-1} \) can be rewritten as \( (2^2)^{n-1} = 2^{2(n-1)} = 2^{2n-2} \).
Now let's consider each option based on the common ratio:
1. \( A_n = \frac{1}{8} \cdot 4^{n-1} \) translates to \( A_n = \frac{1}{8} \cdot 2^{2n-2} \).
2. \( A_n = \frac{1}{8} \cdot 2^{n+1} \) scales the first term differently.
3. \( A_n = \frac{1}{8} \cdot 2^{n-1} \) is also consistent and has a different ratio.
4. \( A_n = \frac{1}{8} \cdot 4^{n+1} \) scales the first term even more.
Each option will yield a different sequence given a specific \( n \). Without the exact recursive formula, there is no way to definitively identify the correct choice. However, knowing we start at \( \frac{1}{8} \):
Check the forms based on what the common ratio could be leading up from the options given (particularly the \( 4 \) versus \( 2 \) ratios).
**Final Note:** If the recursive formula and common ratio information were clearer, we could select an option promptly. Assuming the common ratio is critical for finding the proper sequence explicitly, based on general patterns, identify which option seems logical given your provided answers and knowledge of geometric sequences based on their recursive structure.
- The first term of the sequence is \( a_1 = \frac{1}{8} \).
- The common ratio can be inferred from the answers given.
In general, for a geometric sequence, the explicit formula can be expressed as:
\[
A_n = a_1 \cdot r^{n-1}
\]
Where:
- \( A_n \) is the \( n \)-th term,
- \( a_1 \) is the first term of the sequence,
- \( r \) is the common ratio, and
- \( n \) is the term number.
From the response options provided, we need to analyze the forms:
1. \( A_n = \frac{1}{8} \cdot 4^{n-1} \)
2. \( A_n = \frac{1}{8} \cdot 2^{n+1} \)
3. \( A_n = \frac{1}{8} \cdot 2^{n-1} \)
4. \( A_n = \frac{1}{8} \cdot 4^{n+1} \)
To resolve which could be the explicit formula, we look at the bases involved. Notice that \( 4 \) can be expressed as \( 2^2 \). Thus, \( 4^{n-1} \) can be rewritten as \( (2^2)^{n-1} = 2^{2(n-1)} = 2^{2n-2} \).
Now let's consider each option based on the common ratio:
1. \( A_n = \frac{1}{8} \cdot 4^{n-1} \) translates to \( A_n = \frac{1}{8} \cdot 2^{2n-2} \).
2. \( A_n = \frac{1}{8} \cdot 2^{n+1} \) scales the first term differently.
3. \( A_n = \frac{1}{8} \cdot 2^{n-1} \) is also consistent and has a different ratio.
4. \( A_n = \frac{1}{8} \cdot 4^{n+1} \) scales the first term even more.
Each option will yield a different sequence given a specific \( n \). Without the exact recursive formula, there is no way to definitively identify the correct choice. However, knowing we start at \( \frac{1}{8} \):
Check the forms based on what the common ratio could be leading up from the options given (particularly the \( 4 \) versus \( 2 \) ratios).
**Final Note:** If the recursive formula and common ratio information were clearer, we could select an option promptly. Assuming the common ratio is critical for finding the proper sequence explicitly, based on general patterns, identify which option seems logical given your provided answers and knowledge of geometric sequences based on their recursive structure.