A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entire -electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the -sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index of the -sum graphs, where and denote the valency of the vertex in the molecular graph . Aim of this paper is to compute the lower and upper bounds of the general Randić index for the -sum graphs when . We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly .