Answer:
Yes, the given sequence is geometric and the common ratio is [tex]\frac{2}{3}[/tex].
Step-by-step explanation:
The given sequence is
[tex]\frac{1}{3},\frac{2}{9},\frac{4}{27},\frac{8}{81},\frac{16}{243}[/tex]
If the ratio of all two consecutive terms are same, then the sequence is called geometric.
[tex]r=\frac{a_{n+1}}{a_n}[/tex]
[tex]r_1=\frac{\frac{2}{9}}{\frac{1}{3}}=\frac{2}{3}[/tex]
[tex]r_2=\frac{\frac{4}{27}}{\frac{2}{9}}=\frac{2}{3}[/tex]
[tex]r_3=\frac{\frac{8}{81}}{\frac{4}{27}}=\frac{2}{3}[/tex]
[tex]r_4=\frac{\frac{16}{243}}{\frac{8}{81}}=\frac{2}{3}[/tex]
[tex]r_1=r_2=r_3=r_4[/tex]
Therefore the given sequence is geometric and the common ratio is [tex]\frac{2}{3}[/tex].
