Find the sum of the integers between 100 and 200 that are
(i) divisible by 9. (ii) not divisible by 9.

Explanation

(i) The numbers (integers) between 100 and 200 which is divisible by 9 are 108, 117, 126, … 198.
Let n be the number of terms between 100 and 200 which is divisible by 9.
Here, a = 108, d = 117 - 108 = 9 and `a_(n)=l=198`
` :' a_(n) =l=a+(n-1)d`
` implies 198=108+(n-1)9`
` implies 90=(n-1)9`
` implies n-1=10`
`implies n=11`
` :. ` Sum of terms between 100 and 200 which is divisible by 9,
` S_(n)=(n)/(2)[2a+(n-1)d]`
` implies S_(11)=(11)/(2)[2(108)+(11-1)9]=(11)/(2)[216+90]`
` " " =(11)/(2)xx306=11xx153=1683 `
Hence, required sum of the integers between 100 and 200 that are divisible by 9 is 1683.
(ii) The sum of the integers between 100 and 200 which is not divisible by 9 = (sum of total number between 100 and 200 ) - (sum of total numbers between 100 and 200 which is divisible by 9). ` " " `... (i)
Here, a = 101, d = 102 - 101 = 1 and `a_(n) = l = 199`
` :' a_(n) =l=a+(n-1)d`
`implies 199=101+(n-1)1`
` implies(n-1)=98impliesn=99`
` :.` Sum of terms between 100 and 200,
` S_(n)=(n)/(2)[2a+(n-1)d]`
` implies S_(9)=(99)/(2)[2(101)+(99-1)1]=(99)/(2)[202+98]`
` " " =(99)/(2)xx300=99xx150=14850 `
From Eq. (i), sum of the integers between 100 and 200 which is not divisible by 9
` =14850-1683 " " ` [from part (i)]
` =13167`
Hence, the required sum is 13167.