Let X , Y be Banach modules over a C ∗ -algebra and let r 1 , … , r n ∈ ℝ be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital C ∗ -algebra: ∑ j = 1 n f ( − r j x j + ∑ 1 ≤ i ≤ n , i ≠ j r i x i ) + 2 ∑ i = 1 n r i f ( x i ) = n f ( ∑ i = 1 n r i x i ) . We show that if ∑ i = 1 n r i ≠ 0 , r i , r j ≠ 0 for some 1 ≤ i < j ≤ n and a mapping f : X → Y satisfies the functional equation mentioned above then the mapping f : X → Y is Cauchy additive. As an application, we investigate homomorphisms in unital C ∗ -algebras.