Question about simplification in summation I am reading a book where the following example is shown.
$$= \sum_{0\le n-j\le n} (a+b(n-j)) $$
$$= \sum_{0\le j\le n} (a+bn-bj) $$
Why is n-j being simplified to j? I don't understand why this is possible? To specify my question, what rule of simplification is being used for the part under the summation sign.

It should be clear that $a + b(n-j) = a +bn - bj$, so it is really just a question of how
$$
\sum_{0\leq n-j \leq n} = \sum_{0\leq j\leq n}
$$
First: I assume that $n$ is a fixed number and the the index variable is $j$. So you just have to convince yourself that when $j$ ranges over an interval such that $0\leq n-j \leq n$, then you get the same as $0\leq j \leq n$. And
$$\begin{align}
&0\leq n-j \leq n \quad &\Rightarrow \\
&0 \geq j-n \geq -n \quad&\Rightarrow \\
&n \geq j \geq 0
\end{align}
$$
