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

a_n = 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

a_n = 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

a_n = 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