Solve a_n = -4a_(n-1) - 4a_(n-2), a_0 = 1, a_1 = 0

Step by step

The question

Question: solve aₙ = -4aₙ₋₁ − 4aₙ₋₂, with a₀ = 1 and a₁ = 0.

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 4 and add to 4: 2 and 2.

Both brackets are the same: one root, −2, twice.

4 The general form

With one root twice, the second part gets an extra n: A times −2 to the n, plus B times n times −2 to the n.

5 Use the start values

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

So A = 1.

n = 1: -2A + -2B = 0.

Put A = 1 in.

Take away −2.

Divide by -2: B = −1.

So aₙ = (−2)ⁿ − n × (−2)ⁿ.

6 Check

Check a₂.

The rule: -4 times 0, minus 4 times 1: −4.

The formula with n = 2 gives −4 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.