Question

Given the geometric sequence:
𝑎
𝑛
=
4
,
7.6
,
14.44
,
27.436
,
…
a
n
​
=4,7.6,14.44,27.436,… what is the exponential equation for this sequence?

(2 points)

○
𝑦
=
4
+
1.9
𝑥
y=4+1.9
x


○
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
y=4(1+

Answer 1

To determine the exponential equation for the given geometric sequence, we first observe the terms of the sequence:

- \( a_1 = 4 \)
- \( a_2 = 7.6 \)
- \( a_3 = 14.44 \)
- \( a_4 = 27.436 \)

This sequence appears to be a geometric sequence, which can be expressed in the form:

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

where \( a \) is the first term and \( r \) is the common ratio.

1. **Identify the first term (\( a \))**:
\[
a = 4
\]

2. **Calculate the common ratio (\( r \))**:
We can find \( r \) by dividing the second term by the first term:
\[
r = \frac{a_2}{a_1} = \frac{7.6}{4} = 1.9
\]

3. **Form the general equation**:
Now we can construct the exponential equation using the values of \( a \) and \( r \):
\[
a_n = 4 \cdot (1.9)^{n-1}
\]

4. **Convert to \( y \) form**:
Since we want the equation in the form \( y = a \cdot r^x \), we replace \( n \) with \( x \):
\[
y = 4 \cdot (1.9)^x
\]

Therefore, the correct exponential equation for this sequence is:

\[
\boxed{y = 4(1.9)^x}
\]

This corresponds to the second option you provided.