Define sequence by

            An = 7An-2 + 6An-3 for n>3

and A0 = 0, A1 = 1, A2 = 17.

Solution:

1. Guess shape => An = r^n

2. Plug in

            An = 7An-2 + 6An-3

ð     r^n = 7r^(n-2) + 6r^(n-3)

ð     Divide by r^(n-3)

ð     r^3 = 7r + 6

ð     r^3 – 7r + 6 = 0

ð     (r-1)(r+2)( r-3) = 0             (By guessing)

3. Characteristic Roots = r1 = -1, r2 = -2, r3 = 3

4. Create general form using the fact that the roots are distinct.

            An = α1r1^n + α2r2^n + α3r3^n

                 = α1(-1)^n + α2(-2)^n + α3(3)^n

5. Use initial values to solve for α1, α2, α3.

Solve:

n = 0  0=A0= α1(-1)^0 + α2(-2)^0 + α3(3)^0

n = 1               1=A1= α1(-1)^1 + α2(-2)^1 + α3(3)^1

n = 2               17=A2= α1(-1)^2 + α2(-2)^2 + α3(3)^2

3 equations, 3 unknowns.

a) 0 = α1 + α2 + α3

b) 1 = -α1 – 2α2 + 3α3

c) 17 = α1 + 4α2 + 9α3

a) => α1 = -α2 – α3

            sub into b

1   =-(-α2 – α3) - 2α2 + 3α3

1   = -α2 + 4α3

α2 = 4α3 – 1

Plug into c)

17 = (-α2 – α3) + 4α2 + 9α3

17 = 3α2 + 8α3

17=3(4α3 -1) + 8α3

17=12α3 – 3 + 8α3

17=20α3 – 3

20 = 20α3

1 = α3

From earlier

α2 = 4α3 – 1

α2 = 4(1) – 1

α2 = 3.

Conclusion, sequence has the closed formula

            An = α1r1^n + α2r2^n + α3r3^n

            = -4(-1)^n + (3)(-2)^n + 1(3)^n

Plug values in to check answer!

n=2: 17 = -4 + 12 + 9