Binomial series and relationship between two consecutive terms

Question

I am reading a French book called Traite du calcul differentiel et integral, vol 1. He set out to explain the binomial series and write things as followed:

The analytic expression is put in the form:$(1-x)^m=1+\dfrac{m}{1}x+\dfrac{m(m+1)}{1\cdot2}x^2+\dfrac{m(m+1)(m+2)}{1\cdot2\cdot3}x^3....+\dfrac{m(m-1)...(m-n+2)}{1\cdot2...\cdot(n-1)}x^{n-1}+\dfrac{m(m-1)...(m-n+1)}{1\cdot2...n}x^n+etc....$

It is shown that the relationship between any two consecutive terms is $\dfrac{m-n+1}{n}$

I don't understand how did he derive the relationship between each two terms as $\dfrac{m-n+1}{n}$. Can you help me to explicate this matter better?

Answer 1

Hint:

So, consecutive terms are $m\choose {n-1}$ and $m\choose n$. Now divide the second by the first.

Remember ${m\choose n}=\dfrac{m!}{(m-n)!n!}$.

Answer 2

A minus sign is missing. If you write out the last coefficient in a bit more detail, it is m(m-1)...(m-n+2)(m-n+1)/(1.2. ... .(n-1)n), so everything except (m-n+1)/n cancels out, except for the missing minus sign.