To find the exponential equation for the given geometric sequence \( a_n = 7, 21, 63, 189, \ldots \), we first need to determine the common ratio of the sequence.
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Find the common ratio (r):
- The second term divided by the first term: \( r = \frac{21}{7} = 3 \)
- The third term divided by the second term: \( r = \frac{63}{21} = 3 \)
- The fourth term divided by the third term: \( r = \frac{189}{63} = 3 \)
The common ratio is \( r = 3 \).
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Write the general term (a_n): The formula for the \( n \)-th term of a geometric sequence can be expressed as: \[ a_n = a_1 \cdot r^{(n-1)} \] where \( a_1 \) is the first term.
In this case, \( a_1 = 7 \) and \( r = 3 \): \[ a_n = 7 \cdot 3^{(n-1)} \]
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Express in the form y = a(b)^x: By letting \( x = n - 1 \), we can shift the index: \[ a_n = 7 \cdot 3^x \]
Thus, the exponential equation for the sequence can be written as: \[ y = 7 \cdot 3^x \]
So, the final answer is: \[ y = 7(3)^x \]