To determine the exponential function that represents the geometric sequence 40, 160, 640, ..., we first observe the pattern of the sequence.
- The first term \(a_1\) is 40.
- The second term \(a_2\) is 160, which is \(40 \times 4\).
- The third term \(a_3\) is 640, which is \(160 \times 4\) (or \(40 \times 16\)).
We can see that each term is multiplied by 4 to get the next term, indicating that the common ratio \(r\) is 4.
The general formula for the \(n\)-th term of a geometric sequence can be expressed as:
\[ a_n = a_1 \cdot r^{n-1} \]
Here, \(a_1\) (the first term) is 40, and the common ratio \(r\) is 4. Therefore, the formula for our sequence becomes:
\[ a_n = 40 \cdot 4^{n-1} \]
This matches the form of the option:
\[ a_n = 40(4)^{n-1} \]
So, the correct response is:
a subscript n baseline equals 40 left parenthesis 4 right parenthesis superscript n minus 1 baseline.