To find the eighth term of a geometric sequence, we use the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
Given:
- The first term [tex]\(a_1 = 2\)[/tex]
- The common ratio [tex]\(r = -2\)[/tex]
- We want to find the eighth term, so [tex]\(n = 8\)[/tex]
Let's substitute these values into the formula and solve step-by-step.
1. Start with the formula:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
2. Substitute the known values:
[tex]\[ a_8 = 2 \cdot (-2)^{(8-1)} \][/tex]
3. Calculate the exponent:
[tex]\[ a_8 = 2 \cdot (-2)^7 \][/tex]
4. Evaluate the power of [tex]\(-2\)[/tex]:
[tex]\[ (-2)^7 = -128 \][/tex]
5. Multiply the first term by this result:
[tex]\[ a_8 = 2 \cdot (-128) \][/tex]
6. Conclude the multiplication:
[tex]\[ a_8 = -256 \][/tex]
Therefore, the eighth term of the geometric sequence is [tex]\(\boxed{-256}\)[/tex].
