Question

Find sum of all odd natural numbers between 100,1000

Answer 1

The sum of all natural numbers is n(n+1)/2

number of nattural numbers between 100 to 1000 are 899

i.e.,1000-100-1=899

Therefore sum = 899(899+1)/2

                         =899(900)/2

                         =899(450)

                          =404550.

The sum of all odd natural numbers

That's the sequence 101, 103, 105, ... 997, 999

That's an arithmetic sequence with a1 = 101

an = 999, and d=2  

an = a1 + (n-1)d

999 = 101 + (n-1)2

898 = 2(n-1)

898 = 2n-2

900 = 2n

450 = n

There fore sum = n(n+1)/2

                          =450(450+1)/2

                          = 225(451)

                          =101475  

Hope this helps.....