Proving the Sum of Finite Combinatorial Geometric Series by Mathematical Induction

This paper presents a rigorous mathematical derivation of the finite sum identity for combinatorial geometric series. By decomposing the series into its infinite generating function and a remainder term, it provides a closed-form expression that relates figurate numbers of order k to a polynomial correction. The findings are particularly significant for accelerating real-time network analytics on FPGAs and optimizing stochastic modeling in high-dimensional environments.