To find the least possible value of [tex]\( r \)[/tex], given that [tex]\( r = 3q + 2 \)[/tex] and [tex]\( q = \frac{1}{3^n} \)[/tex] for [tex]\( n = 1, 2, \)[/tex] or 3, we'll proceed through the following steps for each value of [tex]\( n \)[/tex]:
1. Substitute [tex]\( n = 1 \)[/tex] into the formula for [tex]\( q \)[/tex]:
[tex]\[
q = \frac{1}{3^1} = \frac{1}{3}
\][/tex]
Now plug this value of [tex]\( q \)[/tex] into the formula for [tex]\( r \)[/tex]:
[tex]\[
r = 3 \left( \frac{1}{3} \right) + 2 = 1 + 2 = 3
\][/tex]
2. Substitute [tex]\( n = 2 \)[/tex] into the formula for [tex]\( q \)[/tex]:
[tex]\[
q = \frac{1}{3^2} = \frac{1}{9}
\][/tex]
Now plug this value of [tex]\( q \)[/tex] into the formula for [tex]\( r \)[/tex]:
[tex]\[
r = 3 \left( \frac{1}{9} \right) + 2 = \frac{3}{9} + 2 = \frac{1}{3} + 2 = 2 \frac{1}{3}
\][/tex]
3. Substitute [tex]\( n = 3 \)[/tex] into the formula for [tex]\( q \)[/tex]:
[tex]\[
q = \frac{1}{3^3} = \frac{1}{27}
\][/tex]
Now plug this value of [tex]\( q \)[/tex] into the formula for [tex]\( r \)[/tex]:
[tex]\[
r = 3 \left( \frac{1}{27} \right) + 2 = \frac{3}{27} + 2 = \frac{1}{9} + 2 = 2 \frac{1}{9}
\][/tex]
Now, we need to determine the least value among the calculated values of [tex]\( r \)[/tex]:
[tex]\[
3, \ 2 \frac{1}{3}, \ \text{and} \ 2 \frac{1}{9}
\][/tex]
From these values, the minimum is:
[tex]\[
2 \frac{1}{9}
\][/tex]
Therefore, the least possible value of [tex]\( r \)[/tex] is:
[tex]\[
\boxed{2 \frac{1}{9}}
\][/tex]
So, the answer that corresponds to [tex]\( 2 \frac{1}{9} \)[/tex] is [tex]\( G \)[/tex].
