Question

find the sum of all integers fron 1 to 500 which are multiples of 2 or 5
​

Answer 1

N1: Number of terms from 1 to 500 which are multiples of 2 =250

N1:2,4,6,8............500

N2: Number of terms from 1 to 500 which are multiples of 5 =100

N2:5,10,15.......500

N3: Number of terms from 1 to 500 which are multiples of 10 (both 2 and 5) =50

N3:10,20,30...500

Sum of integers which are multiples of 2 from 1 to 500=  

S  

N1

​  

=  

2

N1

​  

[2a+(N1−1)d]

S  

N1

​  

=  

2

250

​  

[2(2)+(250−1)2]

S  

N1

​  

=125[502]=62750

Sum of integers which are multiples of 5 from 1 to 500=

S  

N2

​  

=  

2

100

​  

[2(5)+(100−1)5]

S  

N2

​  

=50[505]=25250

Sum of integers which are multiples of 10 from 1 to 500=

S  

N3

​  

=  

2

50

​  

[2(10)+(50−1)10]

S  

N3

​  

=25[510]=12750

Hence, the sum of integers from 1 to 500 which are multiples of 2 or 5=  

S  

N1

​  

+S  

N2

​  

−S  

N3

​  

 

=62750+25250−12750

=75250.

Sum =75250

Answer 2

Multiples of 2

2,4,6,....................500

a=2

d=4-2=2

l=500

n=500/2 =250

sn=250/2(2+500). sn=n/2(a+l)

=150(502)

=62750

Multiples of 5

5,10,15,...................500

a=5

l=500

n=500/5=100

sn=100/2(2+500)

=50(505)

=25250

10 multiples are common

10,20,30,...................500

a=10

l=500

n=500/10=50

sn=50/2(10+500)

=25(510)

=12750

n=62750+25250 - 12750

=75750