To find the fourth term of a geometric sequence, we need to use the formula for the nth term of a geometric sequence. The formula is:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
where:
- [tex]\( a_n \)[/tex] is the nth term
- [tex]\( a_1 \)[/tex] is the first term of the sequence
- [tex]\( r \)[/tex] is the common ratio
- [tex]\( n \)[/tex] is the term number we want to find
Given:
- The first term ([tex]\( a_1 \)[/tex]) is 7
- The common ratio ([tex]\( r \)[/tex]) is -4
- We want to find the fourth term ([tex]\( a_4 \)[/tex])
Substitute these values into the formula:
[tex]\[ a_4 = 7 \cdot (-4)^{(4-1)} \][/tex]
[tex]\[ a_4 = 7 \cdot (-4)^3 \][/tex]
Now, calculate [tex]\( (-4)^3 \)[/tex]:
[tex]\[ (-4)^3 = (-4) \cdot (-4) \cdot (-4) \][/tex]
[tex]\[ (-4) \cdot (-4) = 16 \][/tex]
[tex]\[ 16 \cdot (-4) = -64 \][/tex]
So,
[tex]\[ (-4)^3 = -64 \][/tex]
Next, multiply this result by the first term:
[tex]\[ a_4 = 7 \cdot (-64) \][/tex]
[tex]\[ a_4 = -448 \][/tex]
Thus, the fourth term of the geometric sequence is:
[tex]\[ a_4 = -448 \][/tex]
