To find an exponential function that represents the geometric sequence 40, 160, 640, we can start by identifying the pattern in the sequence.
-
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.
-
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: \[ a_n = 40 \cdot 4^{(n-1)} \]
-
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.).
1 answer