Find the solution to the following linear homogeneous recurr

Find the solution to the following linear, homogeneous recurrence with constant coefficients: a_n = 12 a_n-1 -35 a_n-2 for n greaterthanorequalto 2 with initial conditions a_0 = - 1, a_1 = -8. The solution is of the form: a_n = alpha_1 (r_1)^n + alpha_2 (r_2)^n for suitable constants alpha_1, alpha_2, r_1, r_2 with r_1 lessthanorequalto r_2. Find these constants. r_1 = r_2 = alpha_1 = alpha_2 =

Solution

using

an = r^n

a_n-1 = r^(n-1)

a_n-2 = r^(n-2)

r^n = 12*r^(n-1) - 35*r^(n-2)

dividing by r^(n-2)

r^2 = 12*r - 35

r^2 - 12r + 35 = 0

(r - 7)(r - 5) = 0

r1 = 5 & r2 = 7

Now using initial conditions:

an = A1*r1^n + A2*r2^n

A1 = alpha1 & A2 = alpha2

when n= 0, a0 = -1

-1 = A1*r1^0 + A2*r2^0

A1 + A2 = -1

when n = 1, a1 = -8

-8 = A1*r1^1 + A2*r2^1

A1*r1 + A2*r2 = -8

5*A1 + 7*A2 = -8

A1 + A2 = -1

Solving both equations:

A2 = -3/2

alpha2 = -1.5

A1 = alpha1 = 0.5