To find the value of the fourth term in a geometric sequence, we use the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
Given:
- The first term [tex]\(a_1 = 15\)[/tex]
- The common ratio [tex]\(r = \frac{1}{3}\)[/tex]
- We need to find the fourth term, so [tex]\(n = 4\)[/tex]
Substitute the values into the formula:
[tex]\[ a_4 = 15 \cdot \left(\frac{1}{3}\right)^{(4-1)} \][/tex]
[tex]\[ a_4 = 15 \cdot \left(\frac{1}{3}\right)^{3} \][/tex]
[tex]\[ a_4 = 15 \cdot \frac{1}{27} \][/tex]
Now, multiply the terms:
[tex]\[ a_4 = \frac{15}{27} \][/tex]
To express [tex]\(\frac{15}{27}\)[/tex] as a simplified fraction, we find the greatest common divisor (GCD) of the numerator and denominator. The GCD of 15 and 27 is 3.
Divide the numerator and the denominator by their GCD:
[tex]\[ \frac{15}{27} = \frac{15 \div 3}{27 \div 3} = \frac{5}{9} \][/tex]
Therefore, the value of the fourth term in the geometric sequence is [tex]\(\boxed{\frac{5}{9}}\)[/tex].
