To find the next term in the given sequence:
[tex]\[ 90, -\frac{135}{2}, \frac{405}{8}, \ldots \][/tex]
we need to examine the pattern in the terms. The sequence appears to be geometric because its terms follow a specific pattern. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio ([tex]\(r\)[/tex]).
### Step-by-Step Solution:
1. Identify the terms in the sequence:
- First term ([tex]\(a_1\)[/tex]) = 90
- Second term ([tex]\(a_2\)[/tex]) = [tex]\(-\frac{135}{2}\)[/tex]
- Third term ([tex]\(a_3\)[/tex]) = [tex]\(\frac{405}{8}\)[/tex]
2. Calculate the common ratio ([tex]\(r\)[/tex]):
The common ratio is calculated by dividing the second term by the first term:
[tex]\[
r_1 = \frac{a_2}{a_1} = \frac{-\frac{135}{2}}{90}
\][/tex]
Simplifying this:
[tex]\[
r_1 = \frac{-135}{2} \times \frac{1}{90} = \frac{-135}{180} = -\frac{3}{4}
\][/tex]
3. Verify the common ratio with the third term:
To confirm the sequence is geometric, we must check the ratio between the third term and the second term:
[tex]\[
r_2 = \frac{a_3}{a_2} = \frac{\frac{405}{8}}{-\frac{135}{2}}
\][/tex]
Simplifying this:
[tex]\[
r_2 = \frac{405}{8} \times \frac{2}{-135} = \frac{405 \times 2}{8 \times -135} = \frac{810}{-1080} = -\frac{3}{4}
\][/tex]
Since [tex]\(r_1 = r_2 = -\frac{3}{4}\)[/tex], the sequence is indeed geometric with a common ratio [tex]\(r = -\frac{3}{4}\)[/tex].
4. Calculate the next term ([tex]\(a_4\)[/tex]):
To find the fourth term in the sequence, multiply the third term by the common ratio:
[tex]\[
a_4 = a_3 \cdot r = \frac{405}{8} \cdot -\frac{3}{4}
\][/tex]
Simplifying this:
[tex]\[
a_4 = \frac{405 \times -3}{8 \times 4} = \frac{-1215}{32}
\][/tex]
Therefore, the next term in the sequence is:
[tex]\[ -\frac{1215}{32} \][/tex]
So the correct answer is:
[tex]\[ -\frac{1215}{32} \][/tex]
which matches option (D).
