An arithmetic progression (AP) is given as 600 + 650 + 700 + 750 +

The first term is $a = 600$.
The common difference is $d = 650 - 600 = 50$.

For the $30^{th}$ term, $n=30$.

The $30^{th}$ term of the AP is $\boxed{\text{2050}}$.

We know $n=30$, $a=600$, and from part (a), $T_{30} = 2050$.

The sum of the first 30 terms of the AP is $\boxed{\text{39750}}$.

2. The quadratic equation $5x^2 + kx + 20 = 0$ has only one root. Determine the possible values of $k$.

For $5x^2 + kx + 20 = 0$:
$a = 5$
$b = k$
$c = 20$

The discriminant is given by $\Delta = b^2 - 4ac$.
Set the discriminant to zero:

The possible values of $k$ are $\boxed{\text{20 and -20}}$.