summations

Summation

Given a sequence

(a_{n})

, then we define

\sum_{i = 1}^{n} a_{i} := a_{1} + a_{2} + \dots + a_{n}

Constant can be Factored

Suppose

c \in 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

(a_{n})

and

(b_{n})

are sequences, then

\sum_{i = 1}^{n} (a_{i} \pm b_{i}) = \sum_{i = 1}^{n} a_{i} \pm \sum_{i = 1}^{n} b_{i}

TODO

Sum of Consecutive Integers

For any

n \in N_{1}

we have that:

\sum_{k = 1}^{n} k = \frac{n \cdot (n + 1)}{2}