To find the exponential equation for the given geometric sequence \( a_n = 7, 21, 63, 189, \ldots \), we first need to identify the common ratio.
- Find the common ratio \( r \):
- \( r = \frac{21}{7} = 3 \)
- \( r = \frac{63}{21} = 3 \)
- \( r = \frac{189}{63} = 3 \)
Thus, the common ratio \( r \) is 3.
- The first term \( a_1 \) of the sequence is 7.
In a geometric sequence, the \( n \)-th term can be expressed using the formula:
\[ a_n = a_1 \cdot r^{n-1} \]
Substituting the values we identified:
- \( a_1 = 7 \)
- \( r = 3 \)
So the exponential equation for the sequence is:
\[ a_n = 7 \cdot 3^{n-1} \]
This equation describes the \( n \)-th term of the geometric sequence.