it can be done in different ways but i am using here hit and trial as that can be lengthy....
note: non decreasing(for a0,a1) means a0<=a1
consider it for single symbol : <a0>
non decreasing sequence= (0), (1),(2) = 3sequence
total sequence possible = 3( ao can contain 0,1,2)
probability(1symbol) = 3/3 =1 ..............@1
now consider it for two symbol sequence : <a0,a1>
so non decreasing sequence will be : (0,0) , (0,1) , (0,2) ,(1,1) ,(1,2) ,(2,2) =6
total sequences from 3 symbols = 3*3 (a0 can take 3values = 0,1,2 also a1)
probability(2 symbols)= 6/9 =2/3....................@2
similarly for three symbols <a0,a1,a2>
non decreasing sequence= (0,0,0), (0,0,1), (0,0,2), (0,1,1), (0,1,2), (0,2,2) ,(1,1,1), (1,1,2) ,(1,2,2), (2,2,2)=10sequences
total sequences = 3*3*3
probability(3 symbols) = 10/27 ...............@3
checking option A for above case :
(n=1) = (1+3*1 +2)/(2 *3) = 6/6 =1 ...........@1 proved
(n=2) = (2*2+ 3*2 +2) / (2*3*3) = 12/18 = 2/3...........@2 proved
(n=3) = (3*3 +3*3 +2)/(2*3*3*3)=20/54=10/27..........@3 proved.
so answer is option A.