Answer the question :

Question

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.

Answer 1

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