f(n+2)=3f(n+1)-f(n)

Solution for f(n+2)=3f(n+1)-f(n) equation:

Simplifying
f(n + 2) = 3f(n + 1) + -1f(n)

Reorder the terms:
f(2 + n) = 3f(n + 1) + -1f(n)
(2 * f + n * f) = 3f(n + 1) + -1f(n)
(2f + fn) = 3f(n + 1) + -1f(n)

Reorder the terms:
2f + fn = 3f(1 + n) + -1f(n)
2f + fn = (1 * 3f + n * 3f) + -1f(n)
2f + fn = (3f + 3fn) + -1f(n)

Multiply f * n
2f + fn = 3f + 3fn + -1fn

Combine like terms: 3fn + -1fn = 2fn
2f + fn = 3f + 2fn

Solving
2f + fn = 3f + 2fn

Solving for variable 'f'.

Move all terms containing f to the left, all other terms to the right.

Add '-3f' to each side of the equation.
2f + -3f + fn = 3f + -3f + 2fn

Combine like terms: 2f + -3f = -1f
-1f + fn = 3f + -3f + 2fn

Combine like terms: 3f + -3f = 0
-1f + fn = 0 + 2fn
-1f + fn = 2fn

Add '-2fn' to each side of the equation.
-1f + fn + -2fn = 2fn + -2fn

Combine like terms: fn + -2fn = -1fn
-1f + -1fn = 2fn + -2fn

Combine like terms: 2fn + -2fn = 0
-1f + -1fn = 0

Factor out the Greatest Common Factor (GCF), '-1f'.
-1f(1 + n) = 0

Ignore the factor -1.
Subproblem 1
Set the factor 'f' equal to zero and attempt to solve:

Simplifying
f = 0

Solving
f = 0

Move all terms containing f to the left, all other terms to the right.

Simplifying
f = 0
Subproblem 2
Set the factor '(1 + n)' equal to zero and attempt to solve:

Simplifying
1 + n = 0

Solving
1 + n = 0

Move all terms containing f to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + n = 0 + -1

Combine like terms: 1 + -1 = 0
0 + n = 0 + -1
n = 0 + -1

Combine like terms: 0 + -1 = -1
n = -1

Add '-1n' to each side of the equation.
n + -1n = -1 + -1n

Combine like terms: n + -1n = 0
0 = -1 + -1n

Simplifying
0 = -1 + -1n

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
f = {0}