Question
Given the geometric sequence: an= 4, 7.6, 14.44, 27.436 ... what is the exponential equation for this sequence?
Answers
GPT-4o mini
To find the exponential equation for the given geometric sequence \( a_n = 4, 7.6, 14.44, 27.436, \ldots \), we need to determine the first term and the common ratio of the sequence.
1. **Identifying the First Term (\( a_1 \)):**
The first term \( a_1 \) is 4.
2. **Finding the Common Ratio (\( r \)):**
The common ratio can be found by dividing any term by its preceding term. Let's calculate it using the first two terms:
\[
r = \frac{a_2}{a_1} = \frac{7.6}{4} = 1.9
\]
We can verify this calculation by calculating the next ratio:
\[
r = \frac{a_3}{a_2} = \frac{14.44}{7.6} = 1.9
\]
And:
\[
r = \frac{a_4}{a_3} = \frac{27.436}{14.44} = 1.9
\]
Thus, the common ratio \( r = 1.9 \).
3. **Formulating the Exponential Equation:**
The general form of a geometric sequence can be expressed as:
\[
a_n = a_1 \cdot r^{n-1}
\]
Substituting the values of \( a_1 \) and \( r \):
\[
a_n = 4 \cdot (1.9)^{n-1}
\]
Thus, the exponential equation for the sequence is:
\[
\boxed{a_n = 4 \cdot (1.9)^{n-1}}
\]
1. **Identifying the First Term (\( a_1 \)):**
The first term \( a_1 \) is 4.
2. **Finding the Common Ratio (\( r \)):**
The common ratio can be found by dividing any term by its preceding term. Let's calculate it using the first two terms:
\[
r = \frac{a_2}{a_1} = \frac{7.6}{4} = 1.9
\]
We can verify this calculation by calculating the next ratio:
\[
r = \frac{a_3}{a_2} = \frac{14.44}{7.6} = 1.9
\]
And:
\[
r = \frac{a_4}{a_3} = \frac{27.436}{14.44} = 1.9
\]
Thus, the common ratio \( r = 1.9 \).
3. **Formulating the Exponential Equation:**
The general form of a geometric sequence can be expressed as:
\[
a_n = a_1 \cdot r^{n-1}
\]
Substituting the values of \( a_1 \) and \( r \):
\[
a_n = 4 \cdot (1.9)^{n-1}
\]
Thus, the exponential equation for the sequence is:
\[
\boxed{a_n = 4 \cdot (1.9)^{n-1}}
\]