Solve a_n = 9a_(n-1) - 20a_(n-2), a_0 = 2, a_1 = 9

Step by step

The question

Question: solve aₙ = 9aₙ₋₁ − 20aₙ₋₂, with a₀ = 2 and a₁ = 9.

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 20 and add to −9: −4 and −5.

So x = 4 or x = 5.

4 The general form

Each root gives a power that works.

Any mix of them works too: A times 4 to the n, plus B times 5 to the n.

5 Use the start values

n = 0: every power to the 0 is 1.

So A + B = 2.

n = 1: 4A + 5B = 9.

From equation 1: A = 2 − B.

Put it into equation 2.

Open the bracket.

Like terms: B.

Take away 8 on both sides.

And A = 2 minus B, that is 1.

So aₙ = 4ⁿ + 5ⁿ.

6 Check

Check a₂.

The rule: 9 times 9, minus 20 times 2: 41.

The formula with n = 2 gives 41 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.