I do not understand what S_k...(*) means. Maybe this approach is helpful:

s[n_] := Sum[a[i], {i, 1, n}]
ss[n_] := Sum[(n - i + 1) a[i], {i, 1, n}]
(* your equation: *)
eqn[n_] := (n + 1) s[n + 1] == Log[2, n + 2] + ss[n]
nMax = 10;  (* as an example *)
vars = Table[a[n], {n, 1, nMax + 1}];
equations = Table[eqn[n], {n, 0, nMax}];
(* solutions: *)
Solve[equations, vars]

Though not rigorous according to this it seems that $$ a[n]= -\frac{\log(n) - \log(n+1)}{n \log(2)} $$ is the general solution.

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