Question

Which exponential function can be used to show the geometric sequence 40, 160, 640, ...?
(1 point)
• an = 40(120)"-1
• an = 40(4)"-1
• an = 40 + 4(n - 1)
0 an = 4(40)"-1

Answer 1

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

The first term \(a_1\) is \(40\).

To find the common ratio \(r\):
\[
r = \frac{a_2}{a_1} = \frac{160}{40} = 4
\]

Now, we confirm that the common ratio is consistent:
\[
r = \frac{a_3}{a_2} = \frac{640}{160} = 4
\]

Since the first term is \(40\) and the common ratio is \(4\), the general formula for the \(n\)-th term of a geometric sequence can be written as:
\[
a_n = a_1 \cdot r^{(n-1)} = 40 \cdot 4^{(n-1)}
\]

Now, let's check the answer choices:

1. \(a_n = 40(120)^{(n-1)}\) - Incorrect, as the ratio is not \(120\).
2. \(a_n = 40(4)^{(n-1)}\) - Correct.
3. \(a_n = 40 + 4(n - 1)\) - Incorrect, as this represents an arithmetic sequence.
4. \(a_n = 4(40)^{(n-1)}\) - Incorrect, as the base should be \(4\) and it should start from \(40\).

Thus, the correct answer is:
\[
\boxed{a_n = 40(4)^{(n-1)}}
\]