Which exponential function can be used to show the geometric sequence 40, 160, 640, . . . ?(1 point)

Question

Responses

an=4(40)n−1
a subscript n baseline equals 4 left parenthesis 40 right parenthesis superscript n minus 1 baseline

an=40(120)n−1
a subscript n baseline equals 40 left parenthesis 120 right parenthesis superscript n minus 1 baseline

an=40(4)n−1
a subscript n baseline equals 40 left parenthesis 4 right parenthesis superscript n minus 1 baseline

an=40+4(n−1)
a subscript n baseline equals 40 plus 4 left parenthesis n minus 1 right parenthesis

Answer 1

To find the exponential function that represents the geometric sequence \( 40, 160, 640, \ldots \), we first need to identify the first term and the common ratio.

1. The first term \( a_1 \) is \( 40 \).
2. To find the common ratio \( r \), we can divide the second term by the first term: \[ r = \frac{160}{40} = 4 \]

Now, we can use the formula for the \( n \)-th term of a geometric sequence: \[ a_n = a_1 \cdot r^{n-1} \] Substituting the values we found: \[ a_n = 40 \cdot 4^{n-1} \]

Now, looking at the response options:

• \( a_n = 4(40)^{n-1} \) (incorrect)
• \( a_n = 40(120)^{n-1} \) (incorrect)
• \( a_n = 40(4)^{n-1} \) (correct)
• \( a_n = 40 + 4(n-1) \) (incorrect; this represents a linear function, not exponential)

Thus, the correct exponential function that shows the geometric sequence \( 40, 160, 640, \ldots \) is: \[ \boxed{a_n = 40(4)^{n-1}} \]