Calculus add formula to derive new formula I was asked to re-write a formula forward and backward and derive a new formula from it.
Here's the problem:

Write $$S=1+2+3+\cdots+N$$ forward and backward $$\begin{array}{rcrcrcr}
S & = & 1 & + & 2 & +\cdots+ & N\\
S & = & N & + & N-1 & +\cdots+ & 1.\\
\overline{2S}&=&\overline{N+1}&+&\overline{N+1}&+\cdots+&\overline{N+1}
\end{array}$$
  Add vertically. Derive formula $(5.1.4)$ by solving for $S$.

$$\sum_{j=1}^N j=\dfrac{N(N+1)}2.\tag{5.1.4}$$
I'm not too sure where to start. Thanks!

There are $n$ terms, and each of them sum to $n+1$.
If we call our sum $S$, we know that $2S=n(n+1)$
From then on, it's a simple matter of rearranging to derive the formula that you need.
