$$ \sum_{k_1=0}^{N-1}\frac{\binom{N-1}{k_1}(\beta -2)^{N-1-k_1}}{1+k_1+\alpha(N-1-k_1)} $$
where $\alpha \in \mathbb{R}$ and $\alpha >1$ (1.3 say), $\beta \in \mathbb{R}$ and $\beta >2 $, and N is a finite natural number
$$ \sum_{k_1=0}^{N-1}\frac{\binom{N-1}{k_1}(\beta -2)^{N-1-k_1}}{1+k_1+\alpha(N-1-k_1)} $$
where $\alpha \in \mathbb{R}$ and $\alpha >1$ (1.3 say), $\beta \in \mathbb{R}$ and $\beta >2 $, and N is a finite natural number