Question

1. Is the system geometric? If so find common ratio

Answer 1

`\large\boxed{\text{yes,}\ (2)/(3)}`

Explanation:

If it's a geometric sequence, then

`(a_(n+1))/(a_n)=a_(n+1):a_n=constant`

Check:

`(2)/(9):(1)/(3)=(2)/(9)\cdot(3)/(1)=(2)/(3)\\\\(4)/(27):(2)/(9)=(4)/(27)\cdot(9)/(2)=(2)/(3)\\\\(8)/(81):(4)/(27)=(8)/(81)\cdot(27)/(4)=(2)/(3)\\\\(16)/(243):(8)/(81)=(16)/(243)\cdot(81)/(8)=(2)/(3)`