Question

All terms of an ap are natural numbers. The sum of face 9 nine consecutive terms beginning with the largest than 200 and smaller than 220. Is the second term is 12 findfind the common difference.

Answer 1

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