algebra precalculus - Is there a way to quickly know the number of elements on a triangle type?

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?

Answer

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