Answer :
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]
[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]
