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, ... ,500)
Each of the above series form an AP.
Let the number of terms in first series be n1.
So, 500=2+(n1−1)2 [nth term of an AP =a+(n−1)d]]
⇒498=(n1−1)2
⇒n1−1=249
⇒n1=250
Let the number of terms in the second list be n2.
So, 500=5+(n2−1)5⇒495=(n2−1)5
⇒99=(n2−1)⇒n2=100
Let the number of terms in the third list be n3.
So,500=10+(n3−1)10⇒490=(n3−1)10⇒49010=n3−1⇒n3=50
Sum of multiples of 2 or 5 from 1 to 500
= Sum of (2,4,6 … 500) + Sum of (5, 10, ….500) – Sum of (10,20,….500)
=n12[2+500]+n22[5+500]−n32[10+500][∵Sn=n2(a+l)]
=2502×502+1002×505−502×510
=250×251+505×50−25×510=62750+25250−12750
=88000−12750=75250