The set P of all nonzero generalized pentagonal numbers is k-additively unique for multiplicative functions for fixed k ≥ 2. That is, if a multiplicative function f satisfies the condition f(a₁ + a₂ + ··· + a_k) = f(a₁) + f(a₂) + ··· + f(a_k) for arbitrary a₁, a₂, ..., a_k ∈ P, then f is the identity function. The known proof for k = 3 uses the generalized Riemann hypothesis (GRH). Here, we give a proof bypassing GRH.