A linear homogeneous recurrence relation of degree 2 with constant coefficients is an equa...

A linear homogeneous recurrence relation of degree 2 with constant coefficients is an equation of the form

where c1 and c2 are real numbers with c2 ≠ 0. It is not difficult to show (see [Ro07]) that if the equation r2− c1r − c2= 0 has two distinct roots r1and r2, then the sequence {an}is a solution of the linear homogeneous recurrence relation an = c1an−1+ c2an−2 if and only if  for n = 0, 1, 2, …, where C1 and C2 are constants. The values of these constants can be found using the two initial terms of the sequence.

Use mathematical induction to prove Theorem 1.7.