Solution for Section 8.3 Question 2

2. Consider the recurrence relation  a_k = 2 a_k-1 + 15 a_k-2   for all integers k2. To find all the sequences of the form 1, t, t², t³, ..., tⁿ,... which satisfy this recurrence relation you need to solve the characteristic equation of the relation. The characteristic equation of this relation is t² - 2 · t - 15 = 0.

By factoring  t² - 2 · t - 15 = (t - 5) (t + 3), so the two values of t which satisfy the characteristic equation are t = 5 and t = -3.  

Hence the two sequences which satisfy the recurrence relation a_k = 2 a_k-1 + 15 a_k-2   for all integers k2 are
1, 5, 25, 125, ..., 5ⁿ,...  and   1, -3, 9, -27, ..., (-3)ⁿ, ...

Back to Section 8.3