To find the value of the fourth term in a geometric sequence, we will use the general formula for the nth term of a geometric sequence:
[tex]\[ a_n = a_1 \times r^{(n-1)} \][/tex]
Where:
- \( a_n \) is the nth term we want to find.
- \( a_1 \) is the first term of the sequence.
- \( r \) is the common ratio.
- \( n \) is the term number we are finding.
Given \( a_1 = 30 \) and \( r = \frac{1}{2} \), we need to find the fourth term, \( a_4 \).
Here are the steps:
1. Identify the given values:
- \( a_1 = 30 \)
- \( r = \frac{1}{2} \)
- \( n = 4 \)
2. Substitute these values into the formula for \( a_n \):
[tex]\[ a_4 = 30 \times \left( \frac{1}{2} \right)^{(4-1)} \][/tex]
3. Simplify the exponent:
[tex]\[ a_4 = 30 \times \left( \frac{1}{2} \right)^3 \][/tex]
4. Calculate the value of \(\left( \frac{1}{2} \right)^3 \):
[tex]\[ \left( \frac{1}{2} \right)^3 = \frac{1}{8} \][/tex]
5. Multiply the result by \( a_1 \):
[tex]\[ a_4 = 30 \times \frac{1}{8} \][/tex]
6. Perform the multiplication:
[tex]\[ a_4 = 30 \times \frac{1}{8} = 30 \div 8 = 3.75 \][/tex]
Therefore, the value of the fourth term in the geometric sequence is [tex]\( 3.75 \)[/tex].
