Solve the following recurrence relation an 2 4an 1 4an

Solve the following recurrence relation a_n + 2 + 4a_n + 1 + 4a_n = 7, n greaterthanorequalto 0, a_0 = 1, a_1 = 2

the characteristic polynomial is

x^2 + 4x + 4 =7

=> x^2 + 4x -3 =0

D = b^2 -4ac

D = 16 - 4*1*3

x1= -b+sqrtD/2a

x1= -4 +2/2

x1 = -1

x2= -b-sqrtD/2a

x1= -4 -2/2

x1 = -3

hence general solution is

an = C1(-1)^n + C2(-3)^n

now a0=1

therefore

1 = c1 +c2

a1 =2

hence

2 = -c1 -3c2

solving both equation

1 = c1 +c2

2 = -c1 -3c2

--------------------------

=> 3 = -2c2

c2 => -3/2

c1 =>1+3/2

c1 = 5/2

now substitute the values in

an = C1(-1)^n + C2(-3)^n