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,


an = 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,


an = 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, an = 198


Let n be no of terms


Then,


an = 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


5
1