Question

Given the geometric sequence: an= 4, 7.6, 14.44, 27.436 ... what is the exponential equation for this sequence? (2 points) Responses y = 4+1.9x y = 4+1.9x y = 4(1.9)x y = 4(1.9)x y = 4 (3.6)x y = 4 (3.6)x y = 4 (1+3.6)x

Answer 1

To determine the exponential equation for the given geometric sequence \( a_n = 4, 7.6, 14.44, 27.436, \ldots \), we first need to identify the common ratio.

To find the common ratio \( r \):
- From \( a_1 (n=1) = 4 \) to \( a_2 (n=2) = 7.6 \):
\[
r = \frac{a_2}{a_1} = \frac{7.6}{4} = 1.9
\]

- From \( a_2 (n=2) = 7.6 \) to \( a_3 (n=3) = 14.44 \):
\[
r = \frac{a_3}{a_2} = \frac{14.44}{7.6} = 1.9
\]

- From \( a_3 (n=3) = 14.44 \) to \( a_4 (n=4) = 27.436 \):
\[
r = \frac{a_4}{a_3} = \frac{27.436}{14.44} = 1.9
\]

Since we have determined that the sequence has a common ratio \( r = 1.9 \) and the first term \( a = 4 \), we can write the general equation of the geometric sequence as:
\[
a_n = a \cdot r^{n-1} = 4 \cdot (1.9)^{n-1}
\]

If we want to express this in the form \( y = 4(1.9)^x \), where \( x = n-1 \), we can conclude that the exponential equation for the sequence is:

\[
y = 4(1.9)^x
\]

Among the options provided, the correct exponential equation for the sequence is:

**\( y = 4(1.9)^x \)**.