Answer
The proofs are below.
Work Step by Step
a) We know that the force is:
$F=k_tx$
In parallel, the spring constant is additive, so $k_t=k_1+k_2$. Thus:
$F=(k_1+k_2)x$
b) We know that the force is:
$F=k_tx$
In series, we can find $k_t$ to be:
$\frac{1}{k_t}=\frac{1}{k_1}+\frac{1}{k_2}$
$\frac{1}{k_t}=\frac{k_1+k_2}{k_1k_2}$
$k_t=\frac{k_1k_2}{k_1+k_2}$
Thus, the force is:
$F=(\frac{k_1k_2}{k_1+k_2})x$
