Certainly! Let's determine the missing term in the sequence [tex]\(180, 60, 30, 18, \text{(?) }\)[/tex].
Here are the given terms in the sequence:
- Term1: 180
- Term2: 60
- Term3: 30
- Term4: 18
First, let's determine the ratios between consecutive terms:
1. The ratio between the first term (180) and the second term (60):
[tex]\[
\text{Ratio 1} = \frac{180}{60} = 3
\][/tex]
2. The ratio between the second term (60) and the third term (30):
[tex]\[
\text{Ratio 2} = \frac{60}{30} = 2
\][/tex]
3. The ratio between the third term (30) and the fourth term (18):
[tex]\[
\text{Ratio 3} = \frac{30}{18} \approx 1.67 \text{ (or exactly } \frac{5}{3}\text{)}
\][/tex]
We observe the pattern of decreasing ratios. Now, we need to find the missing fifth term using the ratio from the third and fourth terms.
To do this, we'll use Ratio 3 to find Term 5:
[tex]\[
\text{Term 5} = \frac{\text{Term 4}}{\text{Ratio 3}} = \frac{18}{1.67} \approx 10.8
\][/tex]
Therefore, the missing term in the sequence is:
[tex]\[
\boxed{10.8}
\][/tex]
