Can sum of the first few consecutive natural numbers is 500?

Answer 1

No.

The sum of the first n consecutive natural numbers is:

sum = n(n+1)/2

So if the sum is 500, then:

n(n+1)/2 = 500

⇒ n(n+1) = 1000

Which means that 1000 must have a pair of factors with a difference of 1 if it is possible.

The pairs factors of 1000 are:

1 x 1000

2 x 500

4 x 250

5 x 200

8 x 125

10 x 100

20 x 50

25 x 40

No two are separated by 1, thus the sum is impossible.