Binomial Random variable to prove to expression

Question

Consider a binomial random variable $X\sim\operatorname{Binom}(n,\theta)$.

1. Show that $f_X(x+1)=f_X(x)\left(\frac{n-x}{x+1}×\frac{\theta}{1-\theta}\right)$.
2. Show that $f_X(x+1)>f_X(x)$ iff $x<(n+1)\theta-1$.

I can get $f_X(x)$ and $f_X(x+1)$ as below but how to solve question 1 and 2? $$f_X(x)=\binom nx\theta^x(1-\theta)^{n-x}$$ $$f_X(x+1)=\binom n{x+1}\theta^{x+1}(1-\theta)^{n-(x+1)}$$

Answer 1

For part (1), compute $$\frac{f_X(x+1)}{f_X(x)}.$$ For part (2), use the result from part (1) to determine when this ratio exceeds $1$.