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All the terms of an AP(arithmetic progression) are natural numbers. Sum of its first nine terms lies between 200 and 220, second term is 12, then find the common difference.

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As we know formula for nth term in AP  

an   = a1 + ( n - 1  ) d
Here 
a1  =  first number of series  
d  =  Common difference 
And
As given second term is 12 , So 

12 =  a1  + ( 2  - 1 ) d 

a1  + d   =  12                        ------------------------- ( 1 )

And we know formula for sum of n terms in AP 

Sn  = n/2 [ 2a1  + ( n  - 1 ) d ]

Here 
200 to 220  = ​9/2 [ 2a1 + ( 9  - 1 ) d ]
200 to 220  = 9/2 [ 2a1 + 8d ]
200 to 220  = 9/2 2 [ a1 + 4d ]
200 to 220  = 9[ 12 -  d  + 4d ]   ( From equation 1 ) 
200 to 220  = 9[ 12 + 3d ]      
200 to 220  = 9× 3 [ 4+ d ]      
200 to 220  = 27 [ 4+ d ]
 

And given AP have  natural numbers and we know natural numbers are ​The natural numbers from 1 upwards: 1, 2, 3, and so on ... 
And their common difference ( d )  also a natural number.
SO we can our sum as multiple of 27 that lies between 200 to 220 , So


216  = 27 ( 4 + d ) 

4 + d  = 8 

d  = 4 
So, common difference =  4    

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