Find $x_k$ in terms of $n$ and $k,1\le k\le n.$ 
For any positive integer $n$ define the sequence $(x_k)_{k\ge0}$ by $x_0=0,x_1=1$ and $$x_{k+2}=\frac{cx_{k+1}-(n-1)x_k}{k+1},k\ge0.$$
  Fix $n$ and let $c$ be the largest real number such that $x_{n+1}=0.$ Find $x_k$ in terms of $n$ and $k,1\le k\le n.$


I can see that $x_{n+1}$ is a polynomial of degree $n$ in $c.$ So I just need to find those $n$ values of $c$ and choose the largest to find $c$. Some help please!

In one method we put $x_k=x^k $. It gives a second order polynomial if $x_k\neq 0$. Since the equation is linear in x a solution is obtained by a linear combination of the two roots power k.
