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Answer :
Let's solve the question of finding the explicit formula for the given geometric sequence: 115, 690, 4,140, 24,840, ...
### Step-by-step Solution:
1. Identify the First Term:
The first term of the sequence, [tex]\( a_1 \)[/tex], is 115.
2. Find the Common Ratio:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio, [tex]\( r \)[/tex].
[tex]\[
r = \frac{\text{second term}}{\text{first term}} = \frac{690}{115}
\][/tex]
After division, we find that [tex]\( r = 6 \)[/tex].
3. Verify the Common Ratio:
Let's check if this ratio holds for the other terms to confirm it's consistent:
- For the third term:
[tex]\[
\frac{\text{third term}}{\text{second term}} = \frac{4140}{690} = 6
\][/tex]
- For the fourth term:
[tex]\[
\frac{\text{fourth term}}{\text{third term}} = \frac{24840}{4140} = 6
\][/tex]
Since the common ratio is consistently 6 for each pair of consecutive terms, we can conclude that the sequence is indeed geometric with [tex]\( r = 6 \)[/tex].
4. Write the Explicit Formula:
The formula for the [tex]\( n \)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
Substituting the values we found:
[tex]\[
a_n = 115 \cdot 6^{(n-1)}
\][/tex]
So, the explicit formula for the sequence is [tex]\( a_n = 115 \cdot (6)^{(n-1)} \)[/tex]. This corresponds to the third option from the list provided.
### Step-by-step Solution:
1. Identify the First Term:
The first term of the sequence, [tex]\( a_1 \)[/tex], is 115.
2. Find the Common Ratio:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio, [tex]\( r \)[/tex].
[tex]\[
r = \frac{\text{second term}}{\text{first term}} = \frac{690}{115}
\][/tex]
After division, we find that [tex]\( r = 6 \)[/tex].
3. Verify the Common Ratio:
Let's check if this ratio holds for the other terms to confirm it's consistent:
- For the third term:
[tex]\[
\frac{\text{third term}}{\text{second term}} = \frac{4140}{690} = 6
\][/tex]
- For the fourth term:
[tex]\[
\frac{\text{fourth term}}{\text{third term}} = \frac{24840}{4140} = 6
\][/tex]
Since the common ratio is consistently 6 for each pair of consecutive terms, we can conclude that the sequence is indeed geometric with [tex]\( r = 6 \)[/tex].
4. Write the Explicit Formula:
The formula for the [tex]\( n \)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
Substituting the values we found:
[tex]\[
a_n = 115 \cdot 6^{(n-1)}
\][/tex]
So, the explicit formula for the sequence is [tex]\( a_n = 115 \cdot (6)^{(n-1)} \)[/tex]. This corresponds to the third option from the list provided.
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