Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To determine the explicit rule for the geometric sequence, let's break down the information given and follow the logical steps to find the correct expression.
### Given Information:
1. Recursive Formula: [tex]\( a_n = 5 \cdot a_{n-1} \)[/tex]
2. First Term: [tex]\( a_1 = 23 \)[/tex]
### Step-by-Step Solution:
1. Identify the General Form of a Geometric Sequence:
A geometric sequence can be defined recursively by:
[tex]\[ a_n = r \cdot a_{n-1} \][/tex]
Here, [tex]\( r \)[/tex] is the common ratio. In our problem, [tex]\( r = 5 \)[/tex].
2. Determine How to Convert the Recursive Formula to an Explicit Formula:
For a geometric sequence, the explicit formula is generally written as:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
where [tex]\( a_1 \)[/tex] is the first term, and [tex]\( r \)[/tex] is the common ratio.
3. Substitute the Known Values:
- [tex]\( a_1 = 23 \)[/tex]
- [tex]\( r = 5 \)[/tex]
Therefore, the explicit formula becomes:
[tex]\[ a_n = 23 \cdot 5^{n-1} \][/tex]
4. Compare with the Given Options:
- Option A: [tex]\( 23 \cdot 5^{n-1} \)[/tex]
- Option B: [tex]\( 5 \cdot 23^{n+1} \)[/tex]
- Option C: [tex]\( 23 \cdot 5^{n+1} \)[/tex]
- Option D: [tex]\( 5 \cdot 23^{n-1} \)[/tex]
The option that matches our explicit formula is:
- Option A: [tex]\( 23 \cdot 5^{n-1} \)[/tex]
### Final Answer:
The explicit rule for the given geometric sequence is Option A. Hence, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
### Given Information:
1. Recursive Formula: [tex]\( a_n = 5 \cdot a_{n-1} \)[/tex]
2. First Term: [tex]\( a_1 = 23 \)[/tex]
### Step-by-Step Solution:
1. Identify the General Form of a Geometric Sequence:
A geometric sequence can be defined recursively by:
[tex]\[ a_n = r \cdot a_{n-1} \][/tex]
Here, [tex]\( r \)[/tex] is the common ratio. In our problem, [tex]\( r = 5 \)[/tex].
2. Determine How to Convert the Recursive Formula to an Explicit Formula:
For a geometric sequence, the explicit formula is generally written as:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
where [tex]\( a_1 \)[/tex] is the first term, and [tex]\( r \)[/tex] is the common ratio.
3. Substitute the Known Values:
- [tex]\( a_1 = 23 \)[/tex]
- [tex]\( r = 5 \)[/tex]
Therefore, the explicit formula becomes:
[tex]\[ a_n = 23 \cdot 5^{n-1} \][/tex]
4. Compare with the Given Options:
- Option A: [tex]\( 23 \cdot 5^{n-1} \)[/tex]
- Option B: [tex]\( 5 \cdot 23^{n+1} \)[/tex]
- Option C: [tex]\( 23 \cdot 5^{n+1} \)[/tex]
- Option D: [tex]\( 5 \cdot 23^{n-1} \)[/tex]
The option that matches our explicit formula is:
- Option A: [tex]\( 23 \cdot 5^{n-1} \)[/tex]
### Final Answer:
The explicit rule for the given geometric sequence is Option A. Hence, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.