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What is the term for a factorial type operation, but with summation instead of products?

I don't technically know the mathematical term, but imagine:

         X         X X        X X X       X X X X      X X X X X     X X X X X X    X X X X X X X   X X X X X X X X  X X X X X X X X X X X X X X X X X X X 

taking the last row that holds 10 elements, the total will be:

10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1  

witch totals 55

just like there is the factor sign 10! that tells us the same but multiplying the numbers, is there a technical term / function, to by other means calculate the total 55?

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    http://en.wikipedia.org/wiki/Triangular_number2011-10-03

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Yes there is, the triangular numbers, and they can be calculated very easily: $T_n=\frac{n(n+1)}{2}.$ Your particular example is $T_{10}=(10\cdot 11)/2=55$.