Solve the recurrence relation a_n = -2a_n-1 + 15a_n-2, n ≥ 2, given a₀ = 1, a₁ = -1.

Question

x² + 2x - 15, the distinct roots 3 and -5, so a_n = C₁(3^n) + C₂(-5)^n. The initial condition gives a₀ = 1 = C₁ - C₂, a₁ = -1 = 3C₁ - 5C₂. We obtain C₁ = C₂ = 1/2 and so a_n = 1/2(3^n) + 1/2(-5)^n.

My question is how does C₁ = C₂ = 1/2 can some please how do you derive to this answer because I'm confused.Thank you for any help.

Answer 1

There wasa typo in the equations derived from the initial conditions. You should have:

The initial condition gives

a₀ = 1 = C₁ + C₂,

a₁ = -1 = 3C₁ - 5C₂

It then easily follows that

C₁ = C₂ = 1/2

Answer 2

Sorry I still don't get it. Can someone please explain?