To find the exponential equation for the sequence \( a_n = 7, 21, 63, 189, \ldots \), we can start by considering the relationship between the terms in the sequence.
We notice that:
- \( a_1 = 7 \)
- \( a_2 = 21 \)
- \( a_3 = 63 \)
- \( a_4 = 189 \)
We can express these terms in a general form. First, we can calculate the ratios of consecutive terms:
\[ \frac{a_2}{a_1} = \frac{21}{7} = 3 \]
\[ \frac{a_3}{a_2} = \frac{63}{21} = 3 \]
\[ \frac{a_4}{a_3} = \frac{189}{63} = 3 \]
From this, we see that each term is multiplied by 3 to get the next term, suggesting that this is a geometric (exponential) sequence.
We can express \( a_n \) in the form:
\[ a_n = a_1 \cdot r^{n-1} \]
where \( a_1 = 7 \) and the common ratio \( r = 3 \). Thus,
\[ a_n = 7 \cdot 3^{n-1} \]
To express this in the form \( y = a \cdot b^x \), we can set \( x = n - 1 \), which gives us:
\[ y = 7 \cdot 3^x \]
Therefore, the exponential equation for the sequence is:
\[ y = 7 \cdot 3^x \]