Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 5 / b / iii / Solution

The characteristic equation of a linear recurrence isIts distinct roots are $3/4$ and $-1/4$, so $p_n=A(3/4{)}^n+B(-1/4{)}^n$. The initial conditions are $p_0=1$ and $p_1=1/2$, giving $A+B=1$ and $3A-B=2$. Hence $A=3/4$, $B=1/4$, and