Why a^n / n is odd (even) is regular Language ?

Question

How could we construct a FA by finding out a pattern in this language

if we take n is odd then we consider n=1 ,3 ,5 ,7  (here string is in AP so we would be able to find out FA)

 it is okay but it it not said that we should only take n =1 3 5 7 9

we could also take n= 3 ,99 ,79 ,5 ,17,9,13  (this is also odd but now there is no pattern now how can we construct a FA )....

can someone explain this ?

Assume same for if n is even.

Answer 1

a^n where n is odd means..
A language which consists of strings such that.. in each such possible string, the number of a's is odd..
OR
L = {a,aaa,aaaaa,aaaaaaa....}  /* There are infinite such strings in the given language */

now your arguement has a problem..
What does n is odd mean..
n can be 1,3,5,7,9... or n is of the form 2p+1, where p =0,1,2,...
So the language should satisfy any possible such value..

When you say we can take n=3,99,79,..
It is wrong because then n doesnot refer to the set of ALL Odd numbers.. it is a proper subset of ODD numbers..