If $k$ is a constant, $m$, $n$ and $p$ are integers and $m \le n < p$, then
- $\dsum_{r=m}^n (a_r+b_r)=\dsum_{r=m}^n a_r +\dsum_{r=m}^n b_r$
- $\dsum_{r=m}^n k =(n-m+1)k$
- $\dsum_{r=m}^n ka_r = k\dsum_{r=m}^n a_r$
- $\dsum_{r=m}^n a_r+\dsum_{r=n+1}^p a_r =\dsum_{r=m}^p a_r$
Find the value of $\dsum_{k=1}^4(7k+3)$.
$\begin{array}{cll}
& \dsum_{k=1}^4(7k+3) \\
= & \dsum_{k=1}^4 7k+\dsum_{k=1}^4 3 & \text{, by property 1} \\
= & 7 \dsum_{k=1}^4 k + (4-1+1)\times 3 & \text{, by property 3 and 2} \\
= & 7(1+2+3+4)+12 \\
= & 82
\end{array}$
