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 $-1\
-1< r < 0$ and $0.5 < a < 1$
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The ratio of two strictly increasing functionsGiven:
\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
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