To find the 8th term in the given geometric sequence, we start by identifying the necessary components of the sequence formula:
The explicit formula for a geometric sequence is given by:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
where:
- \( a_n \) is the n-th term we want to find,
- \( a_1 \) is the first term of the sequence,
- \( r \) is the common ratio,
- \( n \) is the term number.
From the given formula \( a_n = 6 \cdot 3^{(n-1)} \), we can see:
- The first term, \( a_1 \), is 6.
- The common ratio, \( r \), is 3.
- We need to find the 8th term, so \( n = 8 \).
Next, we substitute the given values into the formula:
[tex]\[
a_8 = 6 \cdot 3^{(8-1)}
\][/tex]
Simplify the exponent:
[tex]\[
8-1 = 7
\][/tex]
So the expression becomes:
[tex]\[
a_8 = 6 \cdot 3^7
\][/tex]
Now, calculate \( 3^7 \):
[tex]\[
3^7 = 2187
\][/tex]
Finally, multiply this result by the first term:
[tex]\[
a_8 = 6 \cdot 2187 = 13122
\][/tex]
Thus, the 8th term in the geometric sequence is:
[tex]\[
a_8 = 13122
\][/tex]
