sum over list of functions

I have a Do loop with i running from 1 to 10. Inside that do loop I define functions

f[a_,b_,c_]:= F[a,b,c]

where the F on the rhs are complicated functions that depend implicitly on the index i. After the loop is finished I thus should have 10 different functions fi. These I need to sum to obtain a new function

G[a_,b_,c_]:= Sum[fi[a,b,c],{i,1,10}]

How do I do this? I tried to write each function f into a listelement. I can then indeed sum the listelement, but I cannot keep the functional dependence. I tried to do

f[[i]][a_,b_,c_]:= F[a,b,c]

but that did not work.