- In a geometric sequence, the \( n \)-th term can be expressed using the formula:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where:
- \( a_n \) is the \( n \)-th term,
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.
For this specific geometric sequence:
- The first term \( a_1 = 5 \)
- The common ratio \( r = 6 \)
Thus, the equation for this geometric sequence is:
\[ a_n = 5 \cdot 6^{(n-1)} \]
- To find the 5th term of the sequence (\( n = 5 \)), we can substitute \( n \) into the equation:
\[ a_5 = 5 \cdot 6^{(5-1)} = 5 \cdot 6^{4} \]
Now, we calculate \( 6^4 \):
\[ 6^4 = 6 \cdot 6 \cdot 6 \cdot 6 = 1296 \]
Then substitute back into the equation:
\[ a_5 = 5 \cdot 1296 = 6480 \]
Therefore, the 5th term of the sequence is 6480.