Is closed form of summation less costly? I imagine finding the closed form of an algorithm should yield fewer steps than the original. But in the case of a summation from 1 to n, I'm not sure if it actually does.

Here's the original sum

Sum(N)
   total = 0
   for x = 1 to N
       total = total + x
   ret total


Here's the closed form

Sum(N)
    ret N*(N-1)/2


I think the argument is that '*' is only a single operation vs many '+' operations, but I cannot imagine there exists an internal representation of multiplication on any machine that does not involve repeated addition.

Please correct me if my above claims are incorrect. My questions are: Does the closed form in this particular case yield fewer steps? Is closed form intended to reduce number of operations on the machine itself, and does it ever do that?
I cannot imagine there exists an internal representation of multiplication on any machine that does not involve repeated addition.


Try being more imaginative.  I'm pretty sure that you were taught at school how to calculate, say, $38\times 144$ without adding $144$ thirty-seven times.