To determine which explicit equation correctly represents the given situation, let's analyze each of the provided equations for potential arithmetic or geometric sequences.
### Equation Analysis
1. Equation 1: [tex]\( a_n = 50n - 47 \)[/tex]
- This equation suggests an arithmetic sequence with a first term and a common difference.
- An arithmetic sequence is described by the formula:
[tex]\[
a_n = a_1 + (n - 1)d
\][/tex]
- If we rewrite this equation to fit this form, we get:
[tex]\[
a_n = 50n - 47
\][/tex]
Here, the first term [tex]\( a_1 \)[/tex] would be:
[tex]\[
a_1 = 50 \cdot 1 - 47 = 3
\][/tex]
And the common difference [tex]\( d \)[/tex] is 50.
2. Equation 2: [tex]\( a_n = 50 \cdot 3^{n-1} \)[/tex]
- This indicates a geometric sequence.
- A geometric sequence is described by the formula:
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
- Rewriting this equation in standard form, we have:
[tex]\[
a_n = 50 \cdot (3^{n-1})
\][/tex]
Here, the first term [tex]\( a_1 \)[/tex] is 50 and the common ratio [tex]\( r \)[/tex] is 3.
3. Equation 3: [tex]\( a_n = 3n + 47 \)[/tex]
- This equation implies another arithmetic sequence.
- In standard arithmetic sequence form:
[tex]\[
a_n = a_1 + (n-1)d
\][/tex]
- Rewriting, we get:
[tex]\[
a_n = 3n + 47
\][/tex]
Here, the first term [tex]\( a_1 \)[/tex] would be:
[tex]\[
a_1 = 3 \cdot 1 + 47 = 50
\][/tex]
And the common difference [tex]\( d \)[/tex] is 3.
4. Equation 4: [tex]\( a_n = 3 \cdot 50^{n-1} \)[/tex]
- This suggests another geometric sequence.
- In geometric sequence form:
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
- Rewriting, we get:
[tex]\[
a_n = 3 \cdot (50^{n-1})
\][/tex]
Here, the first term [tex]\( a_1 \)[/tex] is 3 and the common ratio [tex]\( r \)[/tex] is 50.
### Conclusion
Based on the context provided, each equation can describe a valid sequence depending on the specific requirements of the sequence. Therefore, without additional context, the possible solutions can be:
1. [tex]\( a_n = 50n - 47 \)[/tex]
2. [tex]\( a_n = 50 \cdot 3^{n-1} \)[/tex]
3. [tex]\( a_n = 3n + 47 \)[/tex]
4. [tex]\( a_n = 3 \cdot 50^{n-1} \)[/tex]
These represent different possible arithmetic and geometric sequences.
