How was the index replace?

Question

This is page from The Art of Computer Programming by D.E.Knuth. I am interested in circled and underlined

Why he is just change $n-j$ on $j$ ?

Answer 1

There is no change of index, the limits have been simply rewritten: the inequality $0\leq n-j$ is equivalent to $j\leq n$ and the inequality $n-j\leq n$ is equivalent to $0\leq j$. In other words $$0\leq n-j\leq n\quad \Leftrightarrow\quad 0\leq j\leq n.$$

Answer 2

Here is the referred paragraph:

First, there was a change of index according to the rule $(b)$.

If you consider $i=p(j)=n-j$, then: $$\sum_{0\le i\le n} (a+bi)=\sum_{0\le j\le n} (a+bj)=\sum_{0\le n-j\le n} (a+b(n-j)).$$

Second, the index was simplified: $$0\le n-j\le n \stackrel{\times (-1)}\iff \\ 0\ge j-n\ge -n \iff \\ -n\le j-n\le 0 \stackrel{+n}\iff\\ 0\le j\le n. $$