Here , L = {anbm : n ≠ 2m , n ≥ 0 , m ≥ 0}
Now , To write grammar , first we analyze the language and based on this , we will write the grammar...
Here , We have 2 parts in the language :-
1) n > 2m
if m = 0 then accepted strings will be a,a2,a3,a4 ...
if m = 1 then accepted strings will be a3b , a4b , a5b,...
if m = 2 then accepted strings will be a5b2 , a6b2 , a7b2 , ...
if m = 3 then accepted strings will be a7b3 , a8b3 , ..
if m =4 then accepted strings will be a9b4 , a10b4 ,...
Now ,for strings a,a2,a3,a4 , We can write grammar as S ---> aS | a
Now observe 1st string of m=1,2,3,4,.. Similarly 2nd string of m=1,2,3,4...Similarly for 3rd,4th...strings...There is a pattern , a is incrementing 2 times and b is incrementing 1 time every time...Since pattern is starting with aaab.. So , after analyzing the pattern , we can write grammar for this as S ---> aaaAb ; A---->aaAb | aA
So ,overall grammar for 1st part wil be :-
S ---> aS | aaaAb | a
A ----> aaAb | aA | ɛ
Now ,2nd part of the language wil be :-
2) n < 2m
For this part ,
n=0 , strings will be b,bb,bbb,.....So, Grammar will be S --> bX ; X ---> bX | ɛ
now , m = 1 , accepted string is ab (Total 1 string)
m= 2 , accepted strings are ab2 , a2b2 ,a3 b2 (Total 3 strings)
m= 3 , accepted strings are ab3 , a2b3 , a3 b3 , a4 b3 ,a5b3 (Total 5 strings)
m = 4 , accepted strings are ab4 , a2b4 , a3 b4 , a4b4 , a5b4 , a6 b4 , a7b4 (Total 7 strings)
Now , again there is a pattern ..So, grammar for this , S ---> aBb ; B ---> Bb | aBb | ɛ
So, Now , After combining both parts , Grammar will be :-
S ----> S1 | S2
S1 ---> aS1 | aaaAb | a
A ----> aaAb | aA | ɛ
S2 ---> bX | aBb
X ---> bX | ɛ
B ----> Bb | aBb | ɛ