If 2, 6, p, 54 and q are in continued proportion, find the values of p and q.

2, 6, p, 54 and q are in continued proportion then,

$\Rightarrow \dfrac{2}{6} = \dfrac{6}{p} = \dfrac{p}{54} = \dfrac{54}{q}.\\[1em] \text{Solving, } \dfrac{2}{6} = \dfrac{6}{p} \text{ for p,} \\[1em] \Rightarrow p = \dfrac{6}{2} \times 6 \\[1em] \Rightarrow p = 18. \\[1em] \text{Now solving, } \dfrac{p}{54} = \dfrac{54}{q} \text{ for q,} \\[1em] \Rightarrow q = \dfrac{54}{p} \times 54 \\[1em] \Rightarrow q = \dfrac{54}{18} \times 54 \\[1em] \Rightarrow q = 3 \times 54 \\[1em] \Rightarrow q = 162.$

Hence, the value of p = 18 and q = 162.