Question

A geometric sequence starts with the values an=7,21,63,189 . . .
. What is the exponential equation for this sequence?(1 point)
y=_(_)^x

Answer 1

To find the exponential equation for the given geometric sequence \( a_n = 7, 21, 63, 189, \ldots \), we first need to determine the common ratio of the sequence.

1. **Find the common ratio (r)**:
- The second term divided by the first term: \( r = \frac{21}{7} = 3 \)
- The third term divided by the second term: \( r = \frac{63}{21} = 3 \)
- The fourth term divided by the third term: \( r = \frac{189}{63} = 3 \)

The common ratio is \( r = 3 \).

2. **Write the general term (a_n)**:
The formula for the \( n \)-th term of a geometric sequence can be expressed as:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
where \( a_1 \) is the first term.

In this case, \( a_1 = 7 \) and \( r = 3 \):
\[
a_n = 7 \cdot 3^{(n-1)}
\]

3. **Express in the form y = a(b)^x**:
By letting \( x = n - 1 \), we can shift the index:
\[
a_n = 7 \cdot 3^x
\]

Thus, the exponential equation for the sequence can be written as:
\[
y = 7 \cdot 3^x
\]

So, the final answer is:
\[
y = 7(3)^x
\]