be given. If the characteristic equation

has k distinct solutions r₁, r₂, . . . , r_k, then the only solutions to the recurrence are

where the c_i terms are arbitrary constants. (The value of these c_i can be determined by the initial/base-case conditions, but we generally can stop here to find the theta-notation.)

1) For the recurrence:

t_n = 6 t_n-1 - 11 t_n-2 + 6 t_n-3 for n > 2

t₀ = 2

t₁ = 5

t₂ = 15

a) Obtain the characteristic equation

b) Factor the characteristic equation to find the roots

Theorem B.2 Let r be a root of multiplicity m of the characteristic equation for a homogeneous linear recurrence equation with constant coefficients. Then, the general solution includes the terms

where the c_i terms are arbitrary constants.

2) For the recurrence:

t_n = 7 t_n-1 - 15 t_n-2 + 9 t_n-3 for n > 2

t₀ = 0

t₁ = 1

t₂ = 2

a) Obtain the characteristic equation

b) Factor the characteristic equation to find the roots

Theorem B.3 A nonhomogeneous linear recurrence equation of the form

can be transformed into a homogeneous linear recurrence that has the characteristic equation

where b is a constant and p(n) is a polynomial of degree d. If there is more than one term like bⁿp(n) on the right-hand side, each one contributes a term to the characteristic equation.

3) For the recurrence:

t_n = 2 t_n-1 + 2ⁿ - 1 for n > 0

t₀ = 0

a) Obtain the characteristic equation

b) Factor the characteristic equation to find the roots

c) Use the base cases to get values for the c_i constants.