(i) The number (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.

Then,

a_n = a + (n - 1)d

198 = 108 + (n - 1)9

90 = (n - 1)9

n - 1 = 10

n = 11

now, Sum of this AP

[ as last term is given]

= 11(153)

= 1683

(ii) The sum of the integers between 100 and 200 which is not divisible by 9 = (sum of total numbers between 100 and 200) – (sum of total numbers between 100 and 200 which is divisible by 9.

Let the required sum be S

S = S1 - S2

Where S1 is the sum of AP 101, 102, 103, - - - , 199

And S2 is the sum of AP 108, 117, 126, - - - - , 198

For S1

First term, a = 101

Common difference, d = 199

Let n be no of terms

Then,

a_n = a + (n - 1)d

199 = 101 + (n - 1)1

98 = (n - 1)

n = 99

now, Sum of this AP

[ as last term is given]

= 99(150)

= 14850

For S1

First term, a = 108

Common difference, d = 9

Last term, a_n = 198

Let n be no of terms

Then,

a_n = a + (n - 1)d

198 = 108 + (n - 1)9

10 = (n - 1)

n = 11

now, Sum of this AP

= 11(153)

= 1683

Therefore

S = S1 - S2

= 14850 - 1683

= 13167