To find the exponential equation for the given geometric sequence \( a_n = 7, 21, 63, 189, \ldots \), we first identify the common ratio of the sequence.
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Calculate the common ratio: \[ r = \frac{a_2}{a_1} = \frac{21}{7} = 3 \] \[ r = \frac{a_3}{a_2} = \frac{63}{21} = 3 \] \[ r = \frac{a_4}{a_3} = \frac{189}{63} = 3 \] The common ratio \( r \) is 3.
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Find the general term of the geometric sequence: The general form of a geometric sequence can be expressed as: \[ a_n = a_1 \cdot r^{(n-1)} \] where \( a_1 \) is the first term, and \( r \) is the common ratio.
Here, \( a_1 = 7 \) and \( r = 3 \), so: \[ a_n = 7 \cdot 3^{(n-1)} \]
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Express it in terms of \( y \) and \( x \): If we let \( n = x \), then: \[ y = 7 \cdot 3^{(x-1)} \] This can also be rewritten as: \[ y = 7 \cdot 3^x \cdot 3^{-1} \] \[ y = \frac{7}{3} \cdot 3^x \]
However, the standard form can simply be kept as: \[ y = 7 \cdot 3^{(x-1)} \]
Thus, the exponential equation for this sequence is: \[ y = 7 \cdot 3^{(x-1)} \]
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