Solve a_n = 8a_(n-1) - 16a_(n-2), a_0 = 2, a_1 = 12

Step by step

The question

Question: solve aₙ = 8aₙ₋₁ − 16aₙ₋₂, with a₀ = 2 and a₁ = 12.

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 16 and add to −8: −4 and −4.

Both brackets are the same: one root, 4, twice.

4 The general form

With one root twice, the second part gets an extra n: A times 4 to the n, plus B times n times 4 to the n.

5 Use the start values

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

So A = 2.

n = 1: 4A + 4B = 12.

Put A = 2 in.

Take away 8.

Divide by 4: B = 1.

So aₙ = 2 × 4ⁿ + n × 4ⁿ.

6 Check

Check a₂.

The rule: 8 times 12, minus 16 times 2: 64.

The formula with n = 2 gives 64 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.