Answer :
To find the explicit formula for the given geometric sequence, we need to identify the common ratio and the first term.
The sequence given is: [tex]\(115, 690, 4140, 24840, \ldots\)[/tex]
1. Identify the first term ([tex]\(a_1\)[/tex]):
The first term of the sequence is [tex]\(115\)[/tex].
2. Find the common ratio ([tex]\(r\)[/tex]):
To find the common ratio of a geometric sequence, divide the second term by the first term:
[tex]\[
r = \frac{690}{115} = 6
\][/tex]
3. Form the explicit formula:
The explicit formula for a geometric sequence can be written as:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
Plugging in the values we found, [tex]\(a_1 = 115\)[/tex] and [tex]\(r = 6\)[/tex]:
[tex]\[
a_n = 115 \cdot 6^{(n-1)}
\][/tex]
4. Verify with given options:
Compare the expression we found with the given options:
- Option 1: [tex]\(a_n=115 \cdot\left(\frac{1}{6}\right)^{(n-1)}\)[/tex]
- Option 2: [tex]\(a_n=115 \cdot(-6)^{(n-1)}\)[/tex]
- Option 3: [tex]\(a_n=115 \cdot(5)^{(n-1)}\)[/tex]
- Option 4: [tex]\(a_n=115 \cdot(6)^{(n-1)}\)[/tex]
Our formula matches Option 4.
Therefore, the explicit formula for the sequence is [tex]\(a_n=115 \cdot(6)^{(n-1)}\)[/tex].
The sequence given is: [tex]\(115, 690, 4140, 24840, \ldots\)[/tex]
1. Identify the first term ([tex]\(a_1\)[/tex]):
The first term of the sequence is [tex]\(115\)[/tex].
2. Find the common ratio ([tex]\(r\)[/tex]):
To find the common ratio of a geometric sequence, divide the second term by the first term:
[tex]\[
r = \frac{690}{115} = 6
\][/tex]
3. Form the explicit formula:
The explicit formula for a geometric sequence can be written as:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
Plugging in the values we found, [tex]\(a_1 = 115\)[/tex] and [tex]\(r = 6\)[/tex]:
[tex]\[
a_n = 115 \cdot 6^{(n-1)}
\][/tex]
4. Verify with given options:
Compare the expression we found with the given options:
- Option 1: [tex]\(a_n=115 \cdot\left(\frac{1}{6}\right)^{(n-1)}\)[/tex]
- Option 2: [tex]\(a_n=115 \cdot(-6)^{(n-1)}\)[/tex]
- Option 3: [tex]\(a_n=115 \cdot(5)^{(n-1)}\)[/tex]
- Option 4: [tex]\(a_n=115 \cdot(6)^{(n-1)}\)[/tex]
Our formula matches Option 4.
Therefore, the explicit formula for the sequence is [tex]\(a_n=115 \cdot(6)^{(n-1)}\)[/tex].
