In Basic Concepts of Algebraic Topology by Fred Croom, the homology groups of the $n$-skeleton of the closure of an $(n+1)$-simplex are computed in Theorem 2.9. (The geometric carrier of the complex is homeomorphic to the $n$-sphere).
Exercise 21 at the end of the chapter (Chapter 2) asks one to show that the homology groups of positive dimension of the closure of a simplex are trivial and thus deduce that the homology groups of the $n$-sphere are trivial in dimensions $1,2,...,n-1$.
I'm not sure what the author has in mind for this exercise. Of course, the simplex is contractible (whereas the sphere is not), but the concept of homotopy has not yet been introduced. With the exception of the homology group of dimension equal to that of the simplex, it seems to me that solving the exercise amounts to repeating the computations in Theorem 2.9.