Something like this?
Do[Print[{a[i] = RandomInteger[{1, 10}],
b[i] = RandomInteger[{-10, 10}], c[i] = RandomInteger[{1, 10}],
Subscript[A, i][x_] = a[i] Exp[-(x - b[i])^2/(2 c[i]^2)]}], {i, 10}]
G[x_] = Sum[Subscript[A, i][x], {i, 10}]
Plot[Evaluate[
Join[Table[Subscript[A, i][x], {i, 10}], {G[x]}]], {x, -10, 10}]
