theory of computation

Question

check whether given language is regular or not

 

1)   (aⁿ) ⁿ where n ≥1

2)   ( a^m ) ⁿ where n ≥ 1

3)  w= { (( a ² ) ⁿ) *  (( aⁿ )² )*  } where n ≥ 1

Comments

Let p1=$((a^{2})^{n})^{*}$

P2=$((a^{n})^{2})^{*}$ both p1 and p2 is regular so there concatenation is regular

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In case someone is finding L3 hard to understand:

P2, i.e. $((a^n)^2)$* is actually p1 only. See carefully. Both are (a²ⁿ)*. So now you can see why L3 is regular.

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Is there any method or tricky to determine if a language is regular? Please help me I usually find it hard to solve such problems as pumping lemma is not sufficient to determine.

Answer 1

L1 is not regular we will not have the string in A.P only a,a^4,a^9,......

L2 is regular i.e if m=2 a^2,a^4,a^8,.....

L3 is also regular only i.e p1= (( a ² ) ⁿ) * is regular 

p2=(( aⁿ )² )*  is also regular only

concatenation of two regular is regular only.