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