To find the value of the fourth term in a geometric sequence where the first term [tex]\( a_1 = 15 \)[/tex] and the common ratio [tex]\( r = \frac{1}{3} \)[/tex], we can use the formula for the [tex]\( n \)[/tex]-th term of a geometric sequence:
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
For the fourth term ([tex]\( n = 4 \)[/tex]):
[tex]\[
a_4 = a_1 \cdot r^{4-1} = a_1 \cdot r^3
\][/tex]
Given [tex]\( a_1 = 15 \)[/tex] and [tex]\( r = \frac{1}{3} \)[/tex], we substitute these values into the formula:
[tex]\[
a_4 = 15 \cdot \left( \frac{1}{3} \right)^3
\][/tex]
Next, calculate [tex]\( \left( \frac{1}{3} \right)^3 \)[/tex]:
[tex]\[
\left( \frac{1}{3} \right)^3 = \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} = \frac{1}{27}
\][/tex]
Now multiply this result by the first term:
[tex]\[
a_4 = 15 \cdot \frac{1}{27}
\][/tex]
To simplify the multiplication:
[tex]\[
15 \cdot \frac{1}{27} = \frac{15}{27} = \frac{5}{9}
\][/tex]
Therefore, the fourth term expressed as a fraction is:
[tex]\[
\boxed{\frac{5}{9}}
\][/tex]
