Summation

$\sum$ means to sum the results of a formula for all values given

Basic example

\sum_{i = 1}^{4} i^{2} = 1^{2} + 2^{2} + 3^{2} + 4^{2} = 30

Simplification

$a (b + c) = a b + a c$

In other words, constants inside the summed expression can be pulled outside.

Example

\begin{aligned} \sum_{i = 1}^{4} i^{2} = 30 \\ = \sum_{i = 1}^{4} 3 i^{2} = 3 (1^{2}) + 3 (2^{2}) + 3 (3^{2}) + 3 (4^{2}) \\ = 3 [1^{2} + 2^{2} + 3^{2} + 4^{2}] \\ = 3 [\sum_{i = 1}^{4} i^{2}] \end{aligned}

$a + b = b + a$

In other words we can add the terms in any order

Example

\begin{aligned} \sum_{i = 1}^{4} (i^{2} + 2 i) \\ = (1^{2} + 2 (1)) + (2^{2} + 2 (2)) + (3^{2} + 2 (3)) + (4^{2} + 2 (4)) \\ = (1^{2} + 2^{2} + 3^{2} + 4^{2}) + (2 (1)) + 2 (2)) + 2 (3)) + 2 (4)) \\ = \sum_{i = 1}^{4} i^{2} + \sum_{i = 1}^{4} 2 i \end{aligned}

When summing constants, you can multiply the constant by the number of indices you count.

Example

\begin{aligned} \sum_{k = 1}^{10} 5 \\ = 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 \\ = 10 \cdot 5 \\ = 50 \end{aligned}