I was working on some code today which dealt with numbers.

It was about doing a sum of numbers from 1 + 2 + 3 + 4 + …. + N.

I am pretty sure this is known, but here is how it goes:

**Odd numbers:**

Suppose we are doing a sum of numbers 1 to N where N is an odd number.

Lets consider N to be number 5:

1 + 2 + 3 + 4 + 5 = **15**

In the numbers above, 3 is in the center of the sequence, and 5 x 3 = 15. If N is 7 then 7 x 4 = 28 where 4 is at the center of the sequence.

But what if N is huge number like 12345. Finding the number at the center of the sequence is tough. When N was 5 the number at center of the sequence was 3, i.e. 5/2 + 1/2 = 2.5 + 0.5 = 3. This is also true for 7, 11, 13,… Therefore, when N is 12345:

12345/2 + 1/2 = 6173

So the sum of sequence 1 + 2 + 3 + 4 +….. + 12345 = 12345 x 6173 = **76205685**.

i.e. For any given Number N, where N is odd, the sum of 1 to N is **N x (N/2 + 1/2)**

**Even numbers:**

Consider the same scenario when N is an even number. For eg:

1 + 2 + 3 + 4 + 5 + 6 = **21**

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = **36**

Here when N is 6, 21 = (6 x (6/2)) + (6/2).

i.e. For any given number N, where N is even, the sum of 1 to N is **(N x (N/2)) + (N/2)**

All the above is just an observation, I dont have any proof (yet). It might exist already or I might have studied in school too, but dont remember now 🙂

Excellent observation and its absolutely true !. The sum of first n natural numbers as we already know is n(n+1)/ 2 . Your expressions for both even and odd natural numbers reduce to the same and therefore its correct 🙂