This is from Hoel Probability book. The question is: In calculating binomial probabilities, it is convenient to calculate $f\left(x+1\right)$ from $f\left(x\right)$ by the formula $f\left(x+1\right)=k\left(x\right)f\left(x\right)$, where $k\left(x\right)=\frac{\left[\frac{n-x}{x+1}\right]}{\frac{p}{q}}$. Show that this formula is correct.
Accordingly, I should start from the binomial formula $f\left(x\right)=\frac{n!}{\left(x+1\right)!\left(n-\left(x+1\right)\right)!}p^{x+1}q^{n-\left(x+1\right)}$ but I don't know how to proceed during the expassio of terms. Not sure if this is the right approach to answer the question.
All you need is $(x+1)!=(x+1)x!, (n-x-1)!=\frac {(n-x)!} {(n-x)}$ and $p^{x+1}q^{n-(x+1)}=p^{x}q^{n-x} \frac p q$.