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:
$$\begin{align}
\sum_{r=1}^n(ar+b) &amp;= (a1+b) + (a2+b) + \dots + (an+b)\\
&amp;=(a1 + a2 + \dots + an) + (b + b + \dots + b)\\
&amp;=\sum_{r=1}^nar+\sum_{r=1}^nb
\end{align}$$
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 

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