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SUMMATION


Summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total.

For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses; summation of the first 100 natural numbers may be written as 1 + 2 + 3 + 4 + ⋅⋅⋅ + 99 + 100.

Summation can be described using the symbol Σ, the capital Greek letter sigma.

The sum of the first n natural integers can be denoted as nΣi=1 i.

nΣi=m ai = am + am+1 + am+2 + ... + an-1 + an; where i is the index of summation; ai is an indexed variable representing each term of the sum; m is the lower bound of summation, and n is the upper bound of summation; the index starts from m and is incremented by one for each successive term, stopping when i = n.

12+22+32+...+1002 = 100Σk=1 k2

{ai | ai ∈ ℝ, i ∈ {1,...,N}}, NΣi=1 ai = a1 + a2 + a3 + aN

NΣi=1 ai = NΣk=1 ak

6Σk=3 k3 = 33 + 43 + 53 + 63

6Σi=3 i3 = 33 + 43 + 53 + 63

10Σk=1 1/k = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10

NΣk=1 c ⋅ ak = c ⋅ NΣk=1 ak (c ∈ ℝ)

c ⋅ a1 + c ⋅ a2 + ... + c ⋅ an = c ⋅ (a1 + a2 + ... + an)

NΣk=1 ak = MΣk=1 ak + NΣM+1 ak, M,N ∈ ℕ | M < N

NΣk=1 (ak + bk) = NΣk=1 ak + NΣk=1 bk