Solve a_n = 6a_(n-1) - 9a_(n-2), a_0 = 1, a_1 = 6

Step by step

The question

Question: solve aₙ = 6aₙ₋₁ − 9aₙ₋₂, with a₀ = 1 and a₁ = 6.

1 Everything on one side

Move everything to one side.

Each term changes its sign when it moves.

2 The characteristic equation

Powers are good at this: try aₙ = x to the n, for some number x.

Put x to the n in for aₙ, x to the n minus 1 for aₙ₋₁, and so on.

Divide everything by x to the n minus 2: the powers drop to 2, 1 and 0.

This is the characteristic equation.

Careful!

Watch the signs.

The terms moved to the left, so their signs changed.

3 Solve it

Factor: we need two numbers that multiply to 9 and add to −6: −3 and −3.

Both brackets are the same: one root, 3, twice.

4 The general form

With one root twice, the second part gets an extra n: A times 3 to the n, plus B times n times 3 to the n.

5 Use the start values

n = 0: the n in front makes the B part 0.

So A = 1.

n = 1: 3A + 3B = 6.

Put A = 1 in.

Take away 3.

Divide by 3: B = 1.

So aₙ = 3ⁿ + n × 3ⁿ.

6 Check

Check a₂.

The rule: 6 times 6, minus 9 times 1: 27.

The formula with n = 2 gives 27 too.

Right!

Let's look back.

First: everything on one side.

Then: the characteristic equation.

Then: solve it.

Then: the general form.

Then: use the start values.

Last: check.