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Answer :
To determine which rule represents a geometric sequence, let's first look at the characteristics of a geometric sequence. A geometric sequence follows the formula:
[tex]\[ j(n) = a \cdot r^{(n-1)} \][/tex]
where:
- [tex]\( a \)[/tex] is the first term of the sequence,
- [tex]\( r \)[/tex] is the common ratio.
Let's examine each option one by one to determine which one fits this pattern:
- Option A: [tex]\( j(n) = 125 \cdot \left(\frac{1}{5}\right)^{n-1} \)[/tex]
- This is in the correct geometric sequence form with [tex]\( a = 125 \)[/tex] and [tex]\( r = \frac{1}{5} \)[/tex].
- Option B: [tex]\( j(n) = 625 \cdot \left(\frac{1}{5}\right)^{n-1} \)[/tex]
- This also uses the geometric sequence formula, but it starts with [tex]\( a = 625 \)[/tex], not 125 as needed for the solution present earlier.
- Option C: [tex]\( j(n) = 625 \cdot (5)^{n-1} \)[/tex]
- This is a geometric sequence, but the common ratio [tex]\( r = 5 \)[/tex] does not match what we've identified earlier.
- Option D: [tex]\( j(n) = 625 \cdot \left(\frac{1}{5}\right)^n \)[/tex]
- This formula does not fit the standard geometric sequence formula because it uses [tex]\( n \)[/tex] instead of [tex]\( n-1 \)[/tex].
- Option E: [tex]\( j(n) = 125 + 5n \)[/tex]
- This represents an arithmetic sequence, not a geometric one.
From this analysis, option A: [tex]\( j(n) = 125 \cdot \left(\frac{1}{5}\right)^{n-1} \)[/tex] follows the correct format for the geometric sequence and matches the characteristics given. This rule applies for any [tex]\( n \)[/tex] as described, and so it is the correct answer.
[tex]\[ j(n) = a \cdot r^{(n-1)} \][/tex]
where:
- [tex]\( a \)[/tex] is the first term of the sequence,
- [tex]\( r \)[/tex] is the common ratio.
Let's examine each option one by one to determine which one fits this pattern:
- Option A: [tex]\( j(n) = 125 \cdot \left(\frac{1}{5}\right)^{n-1} \)[/tex]
- This is in the correct geometric sequence form with [tex]\( a = 125 \)[/tex] and [tex]\( r = \frac{1}{5} \)[/tex].
- Option B: [tex]\( j(n) = 625 \cdot \left(\frac{1}{5}\right)^{n-1} \)[/tex]
- This also uses the geometric sequence formula, but it starts with [tex]\( a = 625 \)[/tex], not 125 as needed for the solution present earlier.
- Option C: [tex]\( j(n) = 625 \cdot (5)^{n-1} \)[/tex]
- This is a geometric sequence, but the common ratio [tex]\( r = 5 \)[/tex] does not match what we've identified earlier.
- Option D: [tex]\( j(n) = 625 \cdot \left(\frac{1}{5}\right)^n \)[/tex]
- This formula does not fit the standard geometric sequence formula because it uses [tex]\( n \)[/tex] instead of [tex]\( n-1 \)[/tex].
- Option E: [tex]\( j(n) = 125 + 5n \)[/tex]
- This represents an arithmetic sequence, not a geometric one.
From this analysis, option A: [tex]\( j(n) = 125 \cdot \left(\frac{1}{5}\right)^{n-1} \)[/tex] follows the correct format for the geometric sequence and matches the characteristics given. This rule applies for any [tex]\( n \)[/tex] as described, and so it is the correct answer.
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