Kenneth Rosen Edition 7 Exercise 8.2 Question 20 (Page No. 525)

Question

Find the general form of the solutions of the recurrence relation $a_{n} = 8a_{n−2} − 16a_{n−4}.$

Answer 1

Characteristic equation: $r^{ 4} − 8r^{ 2} + 16 = 0$

Factor to find roots.$r^{ 4} − 8r^{ 2} + 16 = 0$

$(r ^{2} − 4)(r^{ 2} − 4) = 0$

$ (r + 2)(r − 2)(r + 2)(r − 2) = 0$

Our roots are r1 = 2 with multiplicity 2 and r2 = −2 with multiplicity 2.

solution

$a_{n}=(A+Bn)(2)^{n}+(C+Dn)(-2)^{n}$