To find the 8th term of a geometric sequence, we need a few key pieces of information:
1. The first term of the sequence (\(a\)).
2. The common ratio (\(r\)).
3. The position of the term we want to find (\(n\)).
Given:
- The first term (\(a\)) is 9.
- The common ratio (\(r\)) is -3.
- The term we are looking for is the 8th term (\(n = 8\)).
The formula for the \(n\)-th term of a geometric sequence is:
[tex]\[ a_n = a \cdot r^{(n-1)} \][/tex]
Let's plug the given values into the formula:
[tex]\[ a_8 = 9 \cdot (-3)^{(8-1)} \][/tex]
[tex]\[ a_8 = 9 \cdot (-3)^7 \][/tex]
Now we need to evaluate \((-3)^7\):
[tex]\[-3^7 = -3 \times -3 \times -3 \times -3 \times -3 \times -3 \times -3\][/tex]
Since the exponent is an odd number, the result of raising a negative number to an odd power is negative:
[tex]\[ (-3)^7 = -2187 \][/tex]
Now we multiply this result by the first term:
[tex]\[ a_8 = 9 \cdot (-2187) \][/tex]
[tex]\[ a_8 = -19683 \][/tex]
So, the 8th term of the geometric sequence is \(-19683\).
Among the given options, the correct answer is:
- [tex]\(-19,683\)[/tex].
