Sum of Sequence as Summation of Difference of Adjacent Terms

Theorem
Let $n \in \Z_{> 0}$ be a strictly positive integer.

Then:
 * $\ds \sum_{k \mathop = 1}^n a_k = n a_n - \sum_{k \mathop = 1}^{n - 1} k \paren {a_{k + 1} - a_k}$