The asymptotic estimation for two classes of generalized Fibonacci sub-sequences

Since the $\mathrm{F}\mathrm{i}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{c}\mathrm{c}\mathrm{i}$ sequence has good properties, it’s important in theory and applications, such as in combinatorics, cryptography, and so on. In this paper, for the generalized Fibonacci sequence $\left\{W_n(a,b,p,q)\right\}$, by using elementary methods and techniques, we respectively give the asymptotic estimation values of ${\left({\sum }_{k=n}^{\mathrm{\infty }}\frac{1}{W_{mk+l}^d}\right)}^{-1}$ and ${\left({\sum }_{k=n}^{\mathrm{\infty }}\frac{{(-1)}^k}{W_{mk+l}^d}\right)}^{-1}$, which generalize the asymptotic estimation results of Yuan et al. \cite{A14} in 2025.