To determine the rule for the given geometric sequence [tex]\(64, -128, 256, -512, \ldots\)[/tex], let's follow a detailed step-by-step approach.
1. Identify the first term ([tex]\(a_1\)[/tex]):
The first term of the sequence is given as [tex]\(64\)[/tex].
2. Find the common ratio ([tex]\(r\)[/tex]):
To find the common ratio, we need to divide the second term by the first term.
- [tex]\(r = \frac{-128}{64} = -2\)[/tex]
3. Verify the common ratio with subsequent terms:
We should verify that this common ratio is consistent by using the first term and multiplying by the common ratio for the next terms in the sequence:
- Second term: [tex]\(64 \times -2 = -128\)[/tex] (which matches the given second term)
- Third term: [tex]\(64 \times (-2)^2 = 64 \times 4 = 256\)[/tex] (which matches the given third term)
- Fourth term: [tex]\(64 \times (-2)^3 = 64 \times -8 = -512\)[/tex] (which matches the given fourth term)
4. General formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
The general form for the [tex]\(n\)[/tex]-th term [tex]\(a_n\)[/tex] of a geometric sequence is given by:
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
Substituting the first term [tex]\(a_1 = 64\)[/tex] and the common ratio [tex]\(r = -2\)[/tex], we get:
[tex]\[
a_n = 64 \cdot (-2)^{n-1}
\][/tex]
Therefore, the rule for the sequence is:
[tex]\[
a_n = 64 (-2)^{n-1}
\][/tex]
So, the correct rule for the geometric sequence is:
[tex]\[
a_n = 64 (-2)^{n-1}
\][/tex]
The correct answer is:
[tex]\[
\boxed{a_n = 64 (-2)^{n-1}}
\][/tex]
