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

Question

Responses

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

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

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

Answer 1

To identify the correct exponential function for the geometric sequence 40, 160, 640, ..., we first observe the pattern in the sequence.

1. The first term \( a_1 \) is 40.
2. The second term \( a_2 \) is 160, which is \( 40 \times 4 \).
3. The third term \( a_3 \) is 640, which is \( 160 \times 4 \) or \( 40 \times 4^2 \).

This indicates that the common ratio is 4.

In general, the formula for a geometric sequence can be written as: \[ a_n = a_1 \cdot r^{n-1} \] where \( a_1 \) is the first term and \( r \) is the common ratio.

Here, \( a_1 = 40 \) and \( r = 4 \). Thus, plugging these values into the formula gives: \[ a_n = 40 \cdot 4^{n-1} \]

Now we can match this with the provided options:

1. \( a_n = 40 + 4(n-1) \) (Linear, not exponential)
2. \( a_n = 40(4)^{n-1} \) (This matches our formula)
3. \( a_n = 4(40)^{n-1} \) (Not correct)
4. \( a_n = 40(120)^{n-1} \) (Not correct)

The correct answer is: \[ a_n = 40(4)^{n-1} \]

Thus, the correct choice is:

\( a_n = 40(4)^{n-1} \).