Question

What exponential function can be used to show the geometric sequence 40, 160, 640

Answer 1

To find an exponential function that represents the geometric sequence 40, 160, 640, we can start by identifying the pattern in the sequence.

1. **Identify the first term and the common ratio**:
- The first term (\( a_1 \)) is 40.
- To find the common ratio (\( r \)), we divide the second term by the first term:
\[
r = \frac{160}{40} = 4
\]
- We can confirm the common ratio with the third term:
\[
\frac{640}{160} = 4
\]
- So, the common ratio is indeed 4.

2. **Write the general form of the geometric sequence**:
- The \( n \)-th term of a geometric sequence can be expressed as:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
- Substituting the values we have:
\[
a_n = 40 \cdot 4^{(n-1)}
\]

3. **Express it in exponential form**:
- We can rewrite this function to represent the sequence:
\[
a_n = 40 \cdot 4^{n-1}
\]

Thus, the exponential function representing the geometric sequence 40, 160, 640 is:
\[
a_n = 40 \cdot 4^{(n-1)}
\]

This function efficiently describes the sequence, where \( n \) is the term number (1 for the first term, 2 for the second term, etc.).