Finding a formula to sum natural numbers up to $n$ [duplicate]

Possible Duplicate:
What is the term for a factorial type operation, but with summation instead of products?

I got this question in homework:

Find an expression for the sum ‫‪

$\sum k = 1 +\cdots + n‬‬$.

and prove it using an induction.

I'm not even near finding the expression. What I did notice is that if $n$ is (for example) 5 then the sum would be

$5^2 - 4^2 + 3^2 - 2^2 + 1^2$

So the first number is always positive and from there on the sign changes.

Any tips on how do I contintue from this point on, assuming I'm in the right direction?

Thanks!

Hint. $$\begin{array}{cccccccccccc} & 1 & + & 2 & + & 3 & + & 4 & + & \cdots & + & n\\ +& n & + &n-1 & + & n-2 & + & n-3 & + & \cdots & + & 1\\ \hline & n+1 & + &n+1 & + &n+1 &+& n+1 & + & \cdots & + & n+1 \end{array}$$

Hint:

If $S_n$ is your sum $S_n=1+2+3+\ldots+n$

then, it's simple you see that

$S_{n}-S_{n-1}=(1+2+\ldots+n-1+n)-(1+2+\ldots+n-1)=$

$(1-1)+(2-2)+\ldots+(n-1 -(n-1))+n=n$.

Now you can use a trick, you can think $S_n$ is a polynomial of degree 2, so

$S_n=an^2+bn+c$

Putting this in the equation

$S_n-S_{n-1}=n$, you will find $a$, $b$ and $c$ and then, you get your sum or $S_n=an^2+bn+c=1+2+3+\ldots+n$.

This is more general, but @ArturoMagidin ways is the easiest.