Sum of Arithmetic Sequence

Theorem
Let $\left \langle{a_k}\right \rangle$ be an arithmetic progression defined as:
 * $a_k = a + k d$ for $n = 0, 1, 2, \ldots, n - 1$

Then its closed-form expression is:

Proof
We have that:


 * $\displaystyle \sum_{k \mathop = 0}^{n - 1} \left({a + k d}\right) = a + \left({a + d}\right) + \left({a + 2 d}\right) + \cdots + \left({a + \left({n - 1}\right) d}\right)$

Then:

So:

Hence the result.

Historical Note
Doubt has recently been cast on the accuracy of the tale about how supposedly discovered this technique at the age of 8.