Find the sum of the integers between 100 and 200 that are

(i) divisible by 9. (ii) not divisible by 9.

(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