Solve a_n = -a_(n-1) + 6a_(n-2), a_0 = 2, a

Step by step

The question

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

a_n = −a_{n − 1} + 6a_{n − 2}, a₀ = 2, a₁ = −1

1 Everything on one side

Move everything to one side.

Each term changes its sign when it moves.

a_n + a_{n − 1} − 6a_{n − 2} = 0

2 The characteristic equation

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

try a_n = xⁿ

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

xⁿ + x^{n − 1} − 6x^{n − 2} = 0

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

This is the characteristic equation.

x² + x − 6 = 0

÷ x^{n − 2} on both sides

Careful!

Watch the signs.

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

Careful: not x² + x + 6 = 0 ✗

3 Solve it

Factor: we need two numbers that multiply to −6 and add to 1: 3 and −2.

(x + 3)(x − 2) = 0

3 × (−2) = −6, 3 + (−2) = 1

So x = −3 or x = 2.

x = −3 or 2

4 The general form

Each root gives a power that works.

Any mix of them works too: A times −3 to the n, plus B times 2 to the n.

a_n = A × (−3)ⁿ + B × 2ⁿ

5 Use the start values

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

So A + B = 2.

A + B = 2 ①

n = 1: −3A + 2B = −1.

-3A + 2B = −1 ②

From equation 1: A = 2 − B.

A = 2 − B

Put it into equation 2.

-3(2 − B) + 2B = −1

Open the bracket.

−6 + 3B + 2B = −1

Like terms: 5B.

−6 + 5B = −1

Take away −6 on both sides.

5B = 5

Divide by 5: B = 1.

B = 1

And A = 2 minus B, that is 1.

A = 2 − 1 = 1

So aₙ = (−3)ⁿ + 2ⁿ.

a_n = (−3)ⁿ + 2ⁿ

6 Check

Check a₂.

The rule: -1 times −1, plus 6 times 2: 13.

rule: a₂ = -1 × (−1) + 6 × 2 = 13

The formula with n = 2 gives 13 too.

Right!

formula: a₂ = 13

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.

Answer