What's the difference between $\sum_{r=1}^n(ar+b)$ and $\sum_{r=1}^nar+b$

Question

Does $$\sum_{r=1}^n(ar+b)=\sum_{r=1}^nar+b$$ or does $$\sum_{r=1}^n(ar+b)=\sum_{r=1}^nar+\sum_{r=1}^nb$$ If I'm given $u_r=ar+b$ how would I substitute that into $$\sum_{r=1}^nu_r$$ Does that mean $$\sum_{r=1}^nu_r=\sum_{r=1}^n(u_r)=\sum_{r=1}^n(ar+b)$$ or $$\sum_{r=1}^nu_r=\sum_{r=1}^nar+b$$ Also $u_r$ means $f(r)$, right?

Answer 1

You can see the correct answer by writing the sums explicitly: $$\begin{align} \sum_{r=1}^n(ar+b) &= (a1+b) + (a2+b) + \dots + (an+b)\\ &=(a1 + a2 + \dots + an) + (b + b + \dots + b)\\ &=\sum_{r=1}^nar+\sum_{r=1}^nb \end{align}$$

Answer 2

This seems to be based on the meaning of the parentheses, in terms of notation. Yes, when you remove the parentheses, you should distribute the "sigma" to both the term ar AND the term b.

Answer 3

There is an ambiguity in the notation

$$\sum_{k=1}^n ak+b $$

does it means

$$\left(\sum_{k=1}^n ak\right)+b $$

or

$$\sum_{k=1}^n \left( ak+b \right)$$