Can someone help me with sum of this expression:$ (n-j)+(n-j)^2?$

Question

Can someone help me prove this transition?

Answer 1

By the change of index $k=n-j$ in the first sum, one gets the second sum.

Answer 2

Simply change the index. Let $j'=n-j$. Then, as $j$ varies from $1 \to n$, it follows that $j'$ varies from $0$ to $n-1$, because $j' =n-j$. When $j=1$, $j'=n-1$, and when $j=n$, then $j'=0$, so then $j'$ must vary between $0$ and $n-1$.

So, $$ \sum_{j=1}^n ((n-j) + (n-j)^2) = \sum_{j'=0}^{n-1} (j' + j'^2) = \sum_{j=0}^{n-1} (j + j^2) $$

Such a transition is called a change of variable. After a certain experience, these changes of variable are omitted from explanations, which is why you have seen a skip in step in your explanation.