math (Gaussian n! formula)
posted by John .
I'm trying to figure out the generalized form of the Gaussian nfactorial (n!) formula.
I keep seeing n!= n(n+1)/2 but that's not working.
For example: 5!= 120 but 5(6)/2 or 6(5/2) is 15; a far cry from 120. I don't understand what I'm doing wrong; distributing gives (n^2+n)/2 which is incorrect.
Also the example I saw as the supposed true story used 1100 and showed it as 101(100/2) = 5050.
What am I missing here?

I don't know where you got that formula, it is obviously wrong.
2!=2*1 which does not equal 2*3/2 
There must be a confusion somewhere.
Gauss as a small boy in school was told by his teacher to add numbers from 1 to 100, after which he can play outside. While everyone else was still adding away, he went out and played after about 2 minutes. His teacher was about to punish him when he told the teacher he finished!
When asked how he did it, he answered:
Write the numbers 1100 and 1001 one on top of the other and add vertically. We have 100 sums of 101 for two rows, so the sum for one row is 101*100/2=5050.
So probably this 1100 = 5050 formula became the famous Gauss formula, but not for factorial.
If you are looking for an approximation to the factorial function, i.e. f(n)=n! (approximately), use the Stirling formula.
lim n>∞ n!=√(2πn)(n/e)^{n}.
See:
http://en.wikipedia.org/wiki/Stirling%27s_approximation
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