Consider the following geometric sequence:

[tex]\[ 115, 690, 4,140, 24,840, \ldots \][/tex]

What is the explicit formula for this sequence?

A. [tex]\[ a_n = 115 \cdot (-6)^{(n-1)} \][/tex]

B. [tex]\[ a_n = 115 \cdot \left(\frac{1}{6}\right)^{(n-1)} \][/tex]

C. [tex]\[ a_n = 115 \cdot (5)^{(n-1)} \][/tex]

D. [tex]\[ a_n = 115 \cdot (6)^{(n-1)} \][/tex]

To find the explicit formula for the given geometric sequence, we need to identify the first term and the common ratio.

1. Identify the first term:
The sequence starts with 115. So, the first term [tex]$ a_1 $[/tex] is 115.

2. Determine 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].

Let's calculate the common ratio using the first two terms:
- First term ([tex]$ a_1 $[/tex]) = 115
- Second term ([tex]$ a_2 $[/tex]) = 690

To find the common ratio [tex]$ r $[/tex], divide the second term by the first term:
[tex]\[
r = \frac{690}{115} = 6
\][/tex]

It's good practice to verify this ratio with other terms in the sequence. Let's check:
- Third term ([tex]$ a_3 $[/tex]) = 4140
[tex]\[
\text{Ratio: } \frac{4140}{690} = 6
\][/tex]
- Fourth term ([tex]$ a_4 $[/tex]) = 24840
[tex]\[
\text{Ratio: } \frac{24840}{4140} = 6
\][/tex]

Since the ratio is consistent across terms, the common ratio [tex]$ r $[/tex] is 6.

3. Write the explicit formula:
The formula for the nth term of a geometric sequence is given by:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]

Now, substituting the first term and the common ratio into the formula, we get:
[tex]\[
a_n = 115 \cdot 6^{(n-1)}
\][/tex]

Therefore, the explicit formula for this sequence is [tex]$ a_n = 115 \cdot (6)^{(n-1)} $[/tex].