To find the next term in the sequence [tex]\(135, -45, 15, -5\)[/tex], we need to analyze the pattern governing the terms.
1. Identify the first term and the positions of the terms:
- The first term ([tex]\(a_1\)[/tex]) is 135.
2. Analyze the pattern:
- Evaluate the possibility of a common ratio (since arithmetic progression doesn't fit).
- The first term is [tex]\(135\)[/tex].
- Second term is [tex]\(-45\)[/tex]. So the change from the first term could be due to a multiplication by some ratio [tex]\(r\)[/tex].
- Let's consider the multiplication factor that transforms 135 to [tex]\(-45\)[/tex]:
[tex]\[
-45 = 135 \times r \implies r = \frac{-45}{135} = -\frac{1}{3}
\][/tex]
- Check if this ratio holds for the other terms:
[tex]\[
15 = -45 \times -\frac{1}{3} \implies -45 = 135 \times \left(-\frac{1}{3}\right), \quad 15 = -45 \times \left(-\frac{1}{3}\right)
\][/tex]
And:
[tex]\[
-5 = 15 \times -\frac{1}{3}
\][/tex]
- The consistent ratio [tex]\(r = -\frac{1}{3}\)[/tex] fits all given terms.
3. Using the common ratio to find the next term:
- The next term can be calculated using the formula for the nth term of a geometric sequence:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
- Here, [tex]\(n = 5\)[/tex] (we need the fifth term).
- Substitute [tex]\(a_1 = 135\)[/tex] and [tex]\(r = -\frac{1}{3}\)[/tex]:
[tex]\[
a_5 = 135 \times \left(-\frac{1}{3}\right)^{4}
\][/tex]
- Calculate [tex]\(\left(-\frac{1}{3}\right)^4\)[/tex]:
[tex]\[
\left(-\frac{1}{3}\right)^4 = \left(\frac{1}{3}\right)^4 = \frac{1}{81}
\][/tex]
- Use this to find [tex]\(a_5\)[/tex]:
[tex]\[
a_5 = 135 \times \frac{1}{81} = \frac{135}{81} = \frac{15}{9} = \frac{5}{3} \approx 1.6666666666666663
\][/tex]
Thus, the next term in the sequence [tex]\(135, -45, 15, -5\)[/tex] is approximately [tex]\(1.6666666666666663\)[/tex].
