To determine the exponential function that represents the geometric sequence 40, 160, 640, ..., we first need to identify the first term and the common ratio of the sequence.
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Identify the first term (a): The first term of the sequence is \( a = 40 \).
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Determine the common ratio (r): To find the common ratio, we divide the second term by the first term: \[ r = \frac{160}{40} = 4 \] We can confirm this by checking the ratio of the third term to the second term: \[ r = \frac{640}{160} = 4 \] This confirms that the common ratio is indeed \( r = 4 \).
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Write the general form of the exponential function: A geometric sequence can be expressed in the form: \[ a_n = a \cdot r^{n-1} \] where \( a_n \) is the \( n \)-th term, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
Plugging our values into the formula, we get: \[ a_n = 40 \cdot 4^{n-1} \]
Thus, the exponential function representing the given geometric sequence is: \[ a_n = 40 \cdot 4^{n-1} \]
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