Question
Geometric Sequences Quick Check
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Question
Which exponential function can be used to show the geometric sequence 40, 160, 640, . . . ?(1 point)
Responses
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 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
Answers
GPT-4o mini
To find the exponential function that represents the given geometric sequence 40, 160, 640, ..., we need to identify the first term and the common ratio.
The first term \( a_1 \) is 40.
To find the common ratio \( r \), we can divide the second term by the first term:
\[
r = \frac{160}{40} = 4
\]
Now, we can express the nth term \( a_n \) of a geometric sequence using the formula:
\[
a_n = a_1 \cdot r^{n-1}
\]
Substituting \( a_1 = 40 \) and \( r = 4 \):
\[
a_n = 40 \cdot 4^{n-1}
\]
Thus, the correct exponential function that represents the sequence is:
\[
\boxed{a_n = 40(4)^{n-1}}
\]
This corresponds to the response:
**an=40(4)n−1**
The first term \( a_1 \) is 40.
To find the common ratio \( r \), we can divide the second term by the first term:
\[
r = \frac{160}{40} = 4
\]
Now, we can express the nth term \( a_n \) of a geometric sequence using the formula:
\[
a_n = a_1 \cdot r^{n-1}
\]
Substituting \( a_1 = 40 \) and \( r = 4 \):
\[
a_n = 40 \cdot 4^{n-1}
\]
Thus, the correct exponential function that represents the sequence is:
\[
\boxed{a_n = 40(4)^{n-1}}
\]
This corresponds to the response:
**an=40(4)n−1**