To determine the formula for the given geometric sequence [tex]\(5, -15, 45, -135, \ldots\)[/tex], let's analyze the sequence step-by-step:
1. Identify the first term and common ratio:
- The first term of the sequence ([tex]\(a_1\)[/tex]) is 5.
- To find the common ratio ([tex]\(r\)[/tex]), we can divide the second term by the first term: [tex]\(-15 / 5 = -3\)[/tex].
2. Verify the common ratio:
- The first term is [tex]\(5\)[/tex].
- The second term ([tex]\(a_2\)[/tex]) is [tex]\(5 \times (-3) = -15\)[/tex].
- The third term ([tex]\(a_3\)[/tex]) is [tex]\(-15 \times (-3) = 45\)[/tex].
- The fourth term ([tex]\(a_4\)[/tex]) is [tex]\(45 \times (-3) = -135\)[/tex].
3. Formulate the general term of the sequence:
- A geometric sequence is generally represented as [tex]\(a_n = a_1 \times r^{n-1}\)[/tex].
- Substituting the first term ([tex]\(a_1 = 5\)[/tex]) and the common ratio ([tex]\(r = -3\)[/tex]), we get:
[tex]\[
a_n = 5 \times (-3)^{n-1}
\][/tex]
4. Verify the formula with specific terms:
- For [tex]\(n = 1\)[/tex]:
[tex]\[
a_1 = 5 \times (-3)^{1-1} = 5 \times 1 = 5
\][/tex]
- For [tex]\(n = 2\)[/tex]:
[tex]\[
a_2 = 5 \times (-3)^{2-1} = 5 \times (-3) = -15
\][/tex]
- For [tex]\(n = 3\)[/tex]:
[tex]\[
a_3 = 5 \times (-3)^{3-1} = 5 \times 9 = 45
\][/tex]
- For [tex]\(n = 4\)[/tex]:
[tex]\[
a_4 = 5 \times (-3)^{4-1} = 5 \times (-27) = -135
\][/tex]
Given the calculations, the correct formula for the sequence is:
[tex]\[
f(n) = 5(-3)^{n-1}
\][/tex]
Among the options provided, this corresponds to the last option:
[tex]\[
f(n) = 5(-3)^{n-1}
\][/tex]
