Summation

Given a sequence \( \left ( a_{n} \right ) \), then we define
\[
\sum_{i = 1}^{n} a _ i := a_{1} + a_{2} + \ldots + a_{n}
\]

Constant can be Factored

Suppose \( c \in \mathbb{R} \), then \( \sum_{i = 1}^{n} c a_{i} = c \sum_{i = 1}^{n} a_{i} \)

TODO

Sum of Added or Subtracted Sequence is the Addition or Subtraction of Sums

Suppose \( \left ( a_{n} \right ) \) and \( \left ( b_{n} \right ) \) are sequences, then
\[
\sum_{i = 1}^{n} \left ( a_{i} \pm b_{i} \right ) = \sum_{i = 1}^{n} a_{i} \pm \sum_{i = 1}^{n} b_{i}
\]

TODO

Sum of Consecutive Integers

For any \( n \in \mathbb{ N } _ 1 \) we have that:
\[
\sum_{k = 1}^{n} k = \frac{n \cdot \left ( n + 1 \right )}{2}
\]