Final answer:
To solve the given recurrence relation, we use the characteristic equation method. For part (a), the general solution is an = A(4^n) + B(2^n), and for part (b), the general solution is an = (A + Bn)(3^n).
Step-by-step explanation:
To solve the given recurrence relation, we will use the characteristic equation method. For part (a), the characteristic equation is r^2 - 6r + 8 = 0. Factoring this equation, we get (r-4)(r-2) = 0, which gives us the roots r1=4 and r2=2. Therefore, the general solution to the recurrence relation is an = A(4^n) + B(2^n), where A and B are constants determined by the initial conditions. Substituting the initial conditions a0 = 1 and a1 = 0 into the general solution, we can solve for A and B.
For part (b), the characteristic equation is r^2 - 6r + 9 = 0, which factors to (r-3)^2 = 0. This gives us a repeated root r=3. The general solution is an = (A + Bn)(3^n), where A and B are again determined by the initial conditions. Substituting a0 = 1 and a1 = 1 into the general solution, we can solve for A and B.