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

Answer 1

To identify the exponential function that represents the geometric sequence 40, 160, 640, ..., we need to first determine the common ratio.

1. The first term \( a_1 = 40 \).
2. The second term \( a_2 = 160 \).
3. The third term \( a_3 = 640 \).

To find the common ratio \( r \):

• \( r = \frac{a_2}{a_1} = \frac{160}{40} = 4 \)
• \( r = \frac{a_3}{a_2} = \frac{640}{160} = 4 \)

The common ratio is consistent, so the sequence is indeed geometric with \( r = 4 \).

In a geometric sequence, the \( n \)-th term can be expressed as:

\[ a_n = a_1 \cdot r^{n-1} \]

Plugging in the values we have:

• \( a_1 = 40 \)
• \( r = 4 \)

Thus, the formula becomes:

\[ a_n = 40 \cdot 4^{n-1} \]

This corresponds to the option:

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

Therefore, the correct response is:

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