# Summation

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A summation is a form of shorthand used foten.

## Definitions

For $b\ge a$, and a set $c$ (or any other algebraic structure), $\sum_{i=a}^{b}c_i=c_a+c_{a+1}+c_{a+2}...+c_{b-1}+c_{b}$. Note that if $a>b$, then the sum is $0$.

## Rules

• $\sum_{i=a}^{b}f(i)+g(i)=\sum_{i=a}^{b}f(i)+\sum_{i=a}^{b}g(i)$
• $\sum_{i=a}^{b}c\cdot f(i)=c\cdot \sum_{i=a}^{b}f(i)$
• $\sum_{i=1}^{n} i= \frac{n(n+1)}{2}$, and in general $\sum_{i=a}^{b} i= \frac{(b-a+1)(a+b)}{2}$
• $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$
• $\sum_{i=1}^{n} i^3 = \left(\sum_{i=1}^{n} i\right)^2 = \left(\frac{n(n+1)}{2}\right)^2$

## Special Summations

Certain types of summations are different from the common variety, such as cyclic sums, and symmetric sums.