Find the Sum of those integers from 1 to 500 which are multiples of 2 or 5.


Since, multiples of 2 or 5 = Multiple of 2 + Multiple of 5 – Multiple of LCM (2, 5) i.e., 10.


Multiples of 2 or 5 from 1 to 500 = List of multiple of 2 from 1 to 500 + List of multiple of 5 from 1 to 500 - List of multiple of 10 from 1 to 500


= (2, 4, 6, …, 500) + (5, 10, 15, …, 500) - (10, 20, 30, …., 500)


All of these list form an AP.


And


Required sum = sum(2, 4, 6, …., 500) + sum(5, 10, 15, …, 500) - sum(10, 20, 30, …., 500)


Consider series


2, 4, 6, …., 500


First term, a = 2


Common difference, d = 2


Let n be no of terms


an = a + (n - 1)d


500 = 2 + (n - 1)2


498 = (n - 1)2


n - 1 = 249


n = 250


let the sum of this AP be S1 using the formula,



S1 = 125(502)


S1 = 62750 [ eqn 1]


Now, Consider series


5, 10, 15, …., 500


First term, a = 5


Common difference, d = 5


Let n be no of terms


By nth term formula


an = a + (n - 1)d


500 = 5 + (n - 1)


495 = (n - 1)5


n - 1 = 99


n = 100


Let the sum of this AP be S_2 using the formula,



S2 = 50(505)


S2 = 25250 [ eqn 2]


Consider series


10, 20, 30, …., 500


First term, a = 10


Common difference, d = 10


Let n be no of terms


an = a + (n - 1)d


500 = 10 + (n - 1)10


490 = (n - 1)10


n - 1 = 49


n = 50


Let the sum of this AP be S1 using the formula,



S3 = 25(510)


S3 = 12750 [ eqn 3]


So required Sum = S1 + S2 - S3


= 62750 + 25250 - 12750


= 75250


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