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]
Here, we need to find the fourth term [tex]\( a_4 \)[/tex]. Setting [tex]\( n = 4 \)[/tex], we have:
[tex]\[ a_4 = a_1 \cdot r^{4-1} \][/tex]
Substitute the given values into the formula:
[tex]\[ a_4 = 15 \cdot \left( \frac{1}{3} \right)^{3} \][/tex]
First, we calculate [tex]\( \left( \frac{1}{3} \right)^3 \)[/tex]:
[tex]\[ \left( \frac{1}{3} \right)^3 = \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27} \][/tex]
Now, multiply this result by the first term [tex]\( a_1 \)[/tex]:
[tex]\[ a_4 = 15 \cdot \frac{1}{27} \][/tex]
To multiply, we multiply the numerator by 15:
[tex]\[ a_4 = \frac{15}{27} \][/tex]
Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 3:
[tex]\[ \frac{15 \div 3}{27 \div 3} = \frac{5}{9} \][/tex]
Thus, the value of the fourth term in the geometric sequence is:
[tex]\[ \boxed{\frac{5}{9}} \][/tex]
