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

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