Stuck on AP using binomial expansion

Question

If the coefficients of $x^{r-1},x^r,x^{r+1}$ in the binomial expansion of $(1+x)^n$ are in AP,prove $n^2-n(4r+1)+4r^2-2=0$ How to start?

Answer 1

$$\binom n{r-1}-2\binom nr+\binom n{r+1}=0$$

$$\frac{n!}{(n-r+1)!(r-1)!}-2\frac{n!}{(n-r)!r!}+\frac{n!}{(n-r-1)(r+1)!}=0$$

Multiplying by $(n-r+1)!(r+1)!$ and dividing by $n!$, $$r(r+1)-2(n-r+1)(r+1)+(n-r+1)(n-r)=0$$