Question

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

Answer 1

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

1. Find the common ratio \( r \):
- \( r = \frac{21}{7} = 3 \)
- \( r = \frac{63}{21} = 3 \)
- \( r = \frac{189}{63} = 3 \)

Thus, the common ratio \( r \) is 3.

2. The first term \( a_1 \) of the sequence is 7.

In a geometric sequence, the \( n \)-th term can be expressed using the formula:

\[
a_n = a_1 \cdot r^{n-1}
\]

Substituting the values we identified:

- \( a_1 = 7 \)
- \( r = 3 \)

So the exponential equation for the sequence is:

\[
a_n = 7 \cdot 3^{n-1}
\]

This equation describes the \( n \)-th term of the geometric sequence.