To find the value of the fourth term in a geometric sequence, we can use the formula for the [tex]\(n\)[/tex]th term of a geometric sequence, which is given by:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
Here, [tex]\(a_1\)[/tex] is the first term of the sequence, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the term number we are looking for.
Given:
- [tex]\(a_1 = 30\)[/tex]
- [tex]\(r = \frac{1}{2}\)[/tex]
- [tex]\(n = 4\)[/tex]
We substitute these values into the formula:
[tex]\[ a_4 = 30 \cdot \left(\frac{1}{2}\right)^{4-1} \][/tex]
Simplify the exponent:
[tex]\[ a_4 = 30 \cdot \left(\frac{1}{2}\right)^3 \][/tex]
Calculate [tex]\(\left(\frac{1}{2}\right)^3\)[/tex]:
[tex]\[ \left(\frac{1}{2}\right)^3 = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8} \][/tex]
So,
[tex]\[ a_4 = 30 \cdot \frac{1}{8} \][/tex]
Multiply the terms:
[tex]\[ a_4 = \frac{30}{8} = 3.75 \][/tex]
Thus, the value of the fourth term in the geometric sequence is:
[tex]\[ \boxed{3.75} \][/tex]
