Let $n{geq}2$ be an even integer. We investigate that if an odd mapping f : X ${ ightarrow}$ Y satisfies the following equation $2_{n-2}C_{frac{n}{2}-1}rf(sumlimits^n_{j=1}{frac{x_j}{r}});+;{sumlimits_{i_k{in}{0,1} atop {{sum}^n_{k=1};i_k={frac{n}{2}}}};rf(sumlimits^n_{i=1}(-1)^{i_k}{frac{x_i}{r}})=2_{n-2}C_{{frac{n}{2}}-1}sumlimits^n_{i=1}f(x_i),$ then f : X ${ ightarrow}$ Y is additive, where $r{in}R$. We also prove the stability in normed group by using shadowing property and the Hyers-Ulam stability of the functional equation in Banach spaces and in Banach modules over unital C-algebras. As an application, we show that every almost linear bijection h : A ${ ightarrow}$ B of unital $C^*$-algebras A and B is a $C^*$-algebra isomorphism when $h(frac{2^s}{r^s}uy)=h(frac{2^s}{r^s}u)h(y)$ for all unitaries u ${in}$ A, all y ${in}$ A, and s = 0, 1, 2,....