To find the exponential function that represents the geometric sequence 40, 160, 640, ..., we first need to identify the common ratio of the sequence.
- The first term \( a_1 = 40 \).
- The second term \( a_2 = 160 \) can be derived from the first term by multiplying by 4: \( 40 \times 4 = 160 \).
- The third term \( a_3 = 640 \) can be derived from the second term similarly: \( 160 \times 4 = 640 \).
Since we see that each term is multiplied by 4 to get the next term, the common ratio \( r = 4 \).
The general form of a geometric sequence can be represented as:
\[ a_n = a_1 \cdot r^{n-1} \]
Substituting in the values we have:
- \( a_1 = 40 \)
- \( r = 4 \)
Thus, the function is:
\[ a_n = 40 \cdot 4^{n-1} \]
Now, let's match this with the options given:
a. \( a_n = 4(40)^{n-1} \) (this does not match) b. \( a_n = 40(120)^{n-1} \) (this does not match) c. \( a_n = 40 + 4(n-1) \) (this is not exponential) d. \( a_n = 40(4)^{n-1} \) (this matches)
Thus, the correct function is:
\[ \boxed{d. , a_n = 40(4)^{n-1}} \]
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