I have the following problem:
Let $X$ be some (path-connected) topological space. I have to show that for two $f,g\in\pi_{n}(X)$ we have that $(f+g)_{\ast}=f_{\ast}+g_{\ast}$, where $\ast$ denotes the induced homomorphism on the singular homology $H_{n}(X)$.
In the lecture, we have defined $\pi_{n}(X,x_{0})$ using spheres, so $f,g$ are functions $f,g:S^{n}\to X$ such that $f(s_{0})=g(s_{0})=x_{0}$. The group operation $f+g$ is then defined to be the composition
$$S^{n}\to S^{n}\vee S^{n}\to X$$
The first map, lets denote it by $\psi$, collapses the equator $S^{n-1}$ in $S^{n}$ to a point and where the second map takes the first sphere in the wedge sum via $f$ to $X$ and the second sphere via $g$ to $X$.
How to I show the claimed equation on homology?