TITLE: Trouble understanding Big O notation for a sum of n integers
QUESTION [5 upvotes]: This problem is an example in a Discrete Math textbook. How can big-O notation be used to estimate the sum of the first n positive integers?
Solution: Because each of the integers in the sum of the first n positive integers does not exceed n, it follows that
$$1+2+...+n \le n+n+...+n=n^2$$
From this inequality it follows that $1+2+...+n$ is $O(n^2)$.
I do not understand how the sum of $n$'s is equal to $n^2$. Can anyone explain to me what the book used to get this?

REPLY [4 votes]: First, you note that$$1+2+\ldots+n = \frac{n^2+n}{2} $$
Since this is a quadratic polynomial, follows that it is $O(n^2)$.