Given: \begin{equation} f_1(a)=\sum_{i=1}^{k^*1} \left(\begin{array}{c} K \\\ i \\ \end{array} \right) \left(1\frac{1}{ar}\right)^i \end{equation} \begin{equation} f_2(a)=\sum_{i=1}^{k^*1} \left(\begin{array}{c} K\\\ i \end{array} \right) \left(1+\frac{1}{a}\right)^i \end{equation} prove or disprove that \begin{equation} f_3(a)=\frac{f_1(a)}{f_2(a)} \end{equation} is an increasing funtion of $a$, where $1< r < 0$ and $0.5 < a < 1$
