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