To solve for [tex]\( p \)[/tex] and [tex]\( q \)[/tex], given that [tex]\(\frac{1}{3}\)[/tex], [tex]\( p \)[/tex], and [tex]\( q \)[/tex] are in a geometric sequence, we should follow these steps:
1. Identify the initial term ([tex]\(a_1\)[/tex]) and the relationship between terms:
[tex]\[
a_1 = \frac{1}{3}
\][/tex]
2. Understand the properties of a geometric sequence:
In a geometric sequence, each term after the first is found by multiplying the previous term by a common ratio [tex]\(r\)[/tex]. For a sequences given as [tex]\(a, a \cdot r, a \cdot r^2, \ldots\)[/tex]:
[tex]\[
a_1 = \frac{1}{3}
\][/tex]
[tex]\[
p = a_1 \cdot r
\][/tex]
[tex]\[
q = a_1 \cdot r^2
\][/tex]
3. Determine the values of [tex]\(p\)[/tex] and [tex]\(q\)[/tex]:
- Since [tex]\(p\)[/tex] is the second term in the sequence:
[tex]\[
p = \frac{1}{3} \cdot 3 = 1
\][/tex]
- And since [tex]\(q\)[/tex] is the third term in the sequence:
[tex]\[
q = \frac{1}{3} \cdot 3^2 = \frac{1}{3} \cdot 9 = 3
\][/tex]
Therefore, the values of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are:
[tex]\[
p = 1
\][/tex]
[tex]\[
q = 3
\][/tex]
