I'm reading What is Mathematics, on page 12 (arithmetic progression), he gives one example of mathematical induction while trying to prove the concept of arithmetic progression. There's something weird here: he starts by proving that the sum of the first $n$ integers is equal to $\frac{n(n+1)}{2}$ by giving this equation:
$1+2+3+\cdots+n=\frac{n(n+1)}{2}.$
Then he adds $(r+1)$ to both sides:
$1+2+3+\cdots+n+(r+1)=\frac{n(n+1)}{2}+(r+1).$
Then he solves it:
$\frac{(r+1)(r+2)}{2}$
Now it seems he's going to prove the arithmetical progression: He says that this can be ordinarily shown by writing the sum $1+2+3+\cdots+n$ in two forms:
$S_n=1+2+\cdots+(n-1)+n$
And:
$S_n=n+(n-1)+\cdots+2+1$
And he states that on adding, we see that each pair of numbers in the same column yields the sum $n+1$ and, since there are $n$ columsn in all, it follows that:
$2S_n=n(n+1).$
I can't understand why he needs to prove the sum of the first $n$ integers first. Can you help me?
Thanks in advance.
EDIT: I've found a copy of the book on scribd, you can check it here. This link will get you in the page I'm in.
EDIT:
I kinda understand the proofs presented in the book now, but I can't see how they are connected to produce a proof for arithmetic progression, I've read the wikipedia article about arithmetic progression and this $a_n = a_m + (n - m)d$ (or at least something similar) would be more plausible as a proof to arithmetic progression - what you think?