Division of coefficients of terms in a binomial expansion

Question

Studying binomial theorem. The question wants me to find out n and r by giving the ratio of coefficients of consecutive terms. But how do I divide the coefficients. Here is the start of my solution

Answer 1

$$R\equiv \ ^nC_{r-2}:^nC_{r-1}:^nC_r = \frac{n!}{(n-r+2)!(r-2)!}:\frac{n!}{(n-r+1)!(r-1)!}:\frac{n!}{(n-r)!r!}$$ $$R\equiv\frac{1}{(n-r+2)(n-r+1)}:\frac{1}{(n-r+1)(r-1)}:\frac{1}{r(r-1)}$$

Taking the first two terms, $$\frac{1}{(n-r+2)}:\frac{1}{(r-1)} \equiv \frac{r-1}{n-r+2} = \frac35$$

Taking the second and third terms, $$\frac{1}{(n-r+1)}:\frac{1}{r}\equiv\frac{r}{n-r+1}=\frac57$$

Can you proceed?