$p_0=a$, $p_1=b$, $bp_n=p_{n+1}+p_{n-1}$ express $p_n$ in terms of $a,b,n$.
Any help would be appreciated, because you guys are splendid.
$p_0=a$, $p_1=b$, $bp_n=p_{n+1}+p_{n-1}$ express $p_n$ in terms of $a,b,n$.
Any help would be appreciated, because you guys are splendid.
You have a linear three-term recurrence. Rearrange as
$p_{n+1}-bp_n+p_{n-1}=0$
and you obtain the characteristic polynomial $x^2-bx+1=0$. This means that $p_n$ can be expressed as
$p_n=cx_1^n+dx_2^n$
where $x_1,x_2$ are the roots of the quadratic. You can determine $c$ and $d$ in terms of $a$ and $b$ by using the initial conditions and solving the resulting linear equations...