# Summation and properties formulas Σ

## Summation notation

\displaystyle \sum_{i=l}^{n}a_{i}

Let’s see the elements of the sigma notation:

i= index or variable of the sum.

a_{i}= i-th element of the sum.

n= number of terms of the sum.

l= start of i

## Properties of the summation

The properties of the sigma are the following:

1. \displaystyle \sum_{i=1}^{n}ka_{i} = k\sum_{i=1}^{n}a_{i}  k = const
2. \displaystyle \sum_{i=1}^{n}(a_{i} + b_{i}) = \displaystyle\sum_{i=1}^{n}a_{i} + \sum_{i=1}^{n}b_{i}

## The formulas of the summation

Next we will see the summation formulas from a constant to i^{8}.

1. \displaystyle \sum_{i=1}^{n}\text{constant} = n\cdot \text{constant}
2. \displaystyle \sum_{i=1}^{n}i = 1 + 2 + 3 + \dots + n = \cfrac{n(n+1)}{2}
3. \displaystyle \sum_{i=1}^{n}i^{2}= 1^{2} + 2^{2} + 3^{2} + \dots + n^{2} = \cfrac{n(n+1)(2n+1)}{6}
4. \displaystyle \sum_{i=1}^{n}i^{3}= 1^{3} + 2^{3} + 3^{3} + \dots + n^{3} = \cfrac{n^{2}(n+1)^{2}}{4}
5. \displaystyle \sum_{i=1}^{n}i^{4}= 1^{4} + 2^{4} + 3^{4} + \dots + n^{4} = \cfrac{n(n+1)(2n+1)(3n^{2} + 3n - 1)}{30}
6. \displaystyle \sum_{i=1}^{n}i^{5}= 1^{5} + 2^{5} + 3^{5} + \dots + n^{5} = \cfrac{n^{2}(n+1)^{2}(2n^{2} + 2n - 1)}{12}
7. \displaystyle \sum_{i=1}^{n}i^{6}= 1^{6} + 2^{6} + 3^{6} + \dots + n^{6} = \cfrac{n(n+1)(2n+1)(3n^{4} + 6n^{3} - 3n + 1)}{42}
8. \displaystyle \sum_{i=1}^{n}i^{7}= 1^{7} + 2^{7} + 3^{7} + \dots + n^{7} = \cfrac{n^{2}(n+1)^{2}(3n^{4} + 6n^{3} - n^{2} - 4n + 2)}{24}
9. \displaystyle \sum_{i=1}^{n}i^{8}= 1^{8} + 2^{8} + 3^{8} + \dots + n^{8} = \cfrac{n(n+1)(2n+1)(5n^{6} + 15n^{5} + 5n^{4} - 15n^{3} - n^{2} + 9n - 3)}{90}

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