Show that the Beta prior is conjugate to a negative binomial likelihood, i.e., if $\mathbf{X} | \theta \sim \mathrm{NegBin}(k,\theta)$ and $\theta \sim \text{Beta}(a, b)$, then $\theta | \mathbf{X} \sim \text{Beta}(a^\ast, b^\ast)$ for a pair of parameters $a^\ast, b^\ast > 0$.
Find an expression for $a^\ast$ and $b^\ast$.
Let us denote the $n$ samples of $X$ as $x_1, \cdots, x_n$.
Note the following:
$$ \begin{align} \pi (\theta | \mathbf{X}) &\propto \frac{ \Gamma(a + b) }{ \Gamma(a) \Gamma(b) } \theta^{a - 1} (1-\theta)^{b - 1} I_{(0 \le \theta \le 1)} \\ &\times \prod_i^n \left[ \frac{\Gamma(k + x_i)}{ \Gamma(k) \Gamma(x_i + 1) } \theta^k (1 - \theta)^{x_i} \right] \\[10pt] &\propto \theta^{a + nk - 1} (1 - \theta)^{b + \sum x_i - 1} I_{(0 \le \theta \le 1)} \end{align} $$
The last line has the form of beta distribution with the parameters $a^\ast = a + nk$, $b^\ast = b + \sum x_i$