$(a)$
\[\begin{align}
\ k_e&=k_1+\frac{1}{\frac{1}{k_2}+\frac{1}{k_3}}\\
&=k_1+\frac{1}{\frac{k_3+k_2}{k_2k_3}}
\end{align}\]
\[\therefore k_e=k_1+\frac{k_2k_3}{k_2+k_3}
\]
$(b)$
\[\begin{align}
k_e&=\frac{1}{\frac{1}{k_1}+\frac{1}{k_2+k_3}}\\&=\frac{1}{\frac{k_2+k_3+k_1}{k_1(k_2+k_3)}}
\end{align}\]
\[
\therefore k_e=\frac{k_1(k_2+k_3)}{k_1+k_2+k_3}
\]
$(c)$
\[
k_e=k_1+k_2
\]
\[
\therefore k_e=k_1+k_2
\]