The primary purpose of this paper is to derive a generalized $(n-k)$ inner product with $n\geq 2$, from the generalized $n$-inner product, which is a generalization of the definition of Misiak [3] of the $n$-inner product for each $k\in\{1,2,...,n-1\}$ and also provide results related to the $n$-normed product induced by generalized $n$-inner product. SAMO NASLOV