Question

Find the solution to the following recurrence:

an=2an−1+8an−2+81n for n≥2 with initial conditions a0=−27,a1=−33
.
an=

Answer 1

To find the solution to the given recurrence relation, use the method of characteristic roots. The solution is of the form an = A(4^n) + B(-2^n) + 81n. Substitute the initial conditions to determine the values of A and B.

Step-by-step explanation:

To find the solution to the given recurrence relation, we'll use the method of characteristic roots. The characteristic equation for the recurrence relation is given by:
r^2 - 2r - 8 = 0

Solving this quadratic equation, we find the roots to be r = 4 and r = -2. So, the general solution for the recurrence relation is:

an = A(4^n) + B(-2^n) + 81n

Using the initial conditions, we can determine the values of A and B:

a0 = -27 corresponds to n = 0, so substituting these values in the general solution:
-27 = A(4^0) + B(-2^0) + 0

From this, we get A - B = -27. Similarly, using the second initial condition, a1 = -33 corresponds to n = 1, we can substitute the values and solve for A and B:

-33 = A(4^1) + B(-2^1) + 81(1)

From this, we get 4A - 2B + 81 = -33. Solving these two equations simultaneously will give us the values of A and B.