If A(n)=1/(n(n+1)) for all positive integers n, what is the

So I'm working throu Manhattan GMAT advanced quant book and got stuck on this problem:

If A(n)=1/(n(n+1)) for all positive integers n, what is the sum of the first 100 elements of An?

The book calculates it by deriving a formula based on pattern by calculating several members of the series and examining changes in the numerator and denominator of the sum of each member + each previous member of the series. That derives to SUM=n/(n+1)

I tried it by using the average formula, AVE=SUM/N and I get a different result. Here're my calculations:

Ave=(A(1)+A(100))/2

A(1)=1/2
A(100)=1/(100(101))=1/10100
Ave=(1/2+1/10100)/2, which ends up being 5051/20200 (unless I made a mistake somewhere along the line).

From there on, based on the Ave formula:
N=100 (first 100 elements of the series)
SUM=AVE*N=5051*100/20200=5051/202

That, clearly doesn't match the answer derived by the method presented in the book!

So did I make mistake in calculations or am I just missing something here?

PS. I searched for this question on the board/google, and nothing came up, but still my apologies if I double-posted something that been discussed previously.