To determine the exponential equation for the given geometric sequence, we first observe the terms of the sequence:
- \( a_1 = 4 \)
- \( a_2 = 7.6 \)
- \( a_3 = 14.44 \)
- \( a_4 = 27.436 \)
This sequence appears to be a geometric sequence, which can be expressed in the form:
\[ a_n = a \cdot r^{n-1} \]
where \( a \) is the first term and \( r \) is the common ratio.
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Identify the first term (\( a \)): \[ a = 4 \]
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Calculate the common ratio (\( r \)): We can find \( r \) by dividing the second term by the first term: \[ r = \frac{a_2}{a_1} = \frac{7.6}{4} = 1.9 \]
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Form the general equation: Now we can construct the exponential equation using the values of \( a \) and \( r \): \[ a_n = 4 \cdot (1.9)^{n-1} \]
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Convert to \( y \) form: Since we want the equation in the form \( y = a \cdot r^x \), we replace \( n \) with \( x \): \[ y = 4 \cdot (1.9)^x \]
Therefore, the correct exponential equation for this sequence is:
\[ \boxed{y = 4(1.9)^x} \]
This corresponds to the second option you provided.