To find the fifth term of a geometric sequence, we use the formula for the \( n \)-th term of a geometric sequence:
[tex]\[
a_n = a \cdot r^{n-1}
\][/tex]
where
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number we want to find.
Given the values:
[tex]\[
a = 7
\][/tex]
[tex]\[
r = -2
\][/tex]
[tex]\[
n = 5
\][/tex]
We substitute these values into the formula for the \( n \)-th term:
[tex]\[
a_5 = 7 \cdot (-2)^{5-1}
\][/tex]
First, we calculate the exponent:
[tex]\[
5 - 1 = 4
\][/tex]
So, we need to find \( (-2)^4 \):
[tex]\[
(-2)^4 = (-2) \cdot (-2) \cdot (-2) \cdot (-2) = 16
\][/tex]
Now we substitute back into our formula:
[tex]\[
a_5 = 7 \cdot 16
\][/tex]
Finally, we perform the multiplication:
[tex]\[
7 \cdot 16 = 112
\][/tex]
Therefore, the fifth term of the geometric sequence is:
[tex]\[
\boxed{112}
\][/tex]
