Whijavascript:void(0)ch exponential function can be used to show the geometric sequence 40, 160, 640, . . . ?(1 point)

Question

Responses

an=4(40)n−1

=
4
(
40
)

−
1
a subscript n baseline equals 4 left parenthesis 40 right parenthesis superscript n minus 1 baseline

an=40(120)n−1

=
40
(
120
)

−
1
a subscript n baseline equals 40 left parenthesis 120 right parenthesis superscript n minus 1 baseline

an=40+4(n−1)

=
40
+
4
(

−
1
)
a subscript n baseline equals 40 plus 4 left parenthesis n minus 1 right parenthesis

an=40(4)n−1

Answer 1

To determine which exponential function represents the geometric sequence \( 40, 160, 640, \ldots \), we first need to analyze the pattern in the sequence.

1. Identify the first term: \( a_1 = 40 \)
2. Identify the common ratio:
   • From \( 40 \) to \( 160 \), the ratio is \( \frac{160}{40} = 4 \).
   • From \( 160 \) to \( 640 \), the ratio is \( \frac{640}{160} = 4 \).
   • The common ratio is \( r = 4 \).

Next, we can express the \( n \)-th term of a geometric sequence with the formula:

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

Substituting the values from our geometric sequence:

• \( a_1 = 40 \)
• \( r = 4 \)

Thus, the formula for the \( n \)-th term becomes:

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

So, out of the given options, this corresponds to:

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

Therefore, the correct exponential function that shows the geometric sequence \( 40, 160, 640, \ldots \) is:

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