Question

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

Answer 1

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

1. **Identify the first term (a)**:
The first term of the sequence is \( a = 40 \).

2. **Determine the common ratio (r)**:
To find the common ratio, we divide the second term by the first term:
\[
r = \frac{160}{40} = 4
\]
We can confirm this by checking the ratio of the third term to the second term:
\[
r = \frac{640}{160} = 4
\]
This confirms that the common ratio is indeed \( r = 4 \).

3. **Write the general form of the exponential function**:
A geometric sequence can be expressed in the form:
\[
a_n = a \cdot r^{n-1}
\]
where \( a_n \) is the \( n \)-th term, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

Plugging our values into the formula, we get:
\[
a_n = 40 \cdot 4^{n-1}
\]

Thus, the exponential function representing the given geometric sequence is:
\[
a_n = 40 \cdot 4^{n-1}
\]