Solve a_n = a_(n-1) + 12a_(n-2), a_0 = 2, a

Step by step

The question

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

a_n = a_{n − 1} + 12a_{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} − 12a_{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} − 12x^{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 − 12 = 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 + 12 = 0 ✗

3 Solve it

Factor: we need two numbers that multiply to −12 and add to −1: 3 and −4.

(x + 3)(x − 4) = 0

3 × (−4) = −12, 3 + (−4) = −1

So x = −3 or x = 4.

x = −3 or 4

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 4 to the n.

a_n = A × (−3)ⁿ + B × 4ⁿ

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 + 4B = 1.

-3A + 4B = 1 ②

From equation 1: A = 2 − B.

A = 2 − B

Put it into equation 2.

-3(2 − B) + 4B = 1

Open the bracket.

−6 + 3B + 4B = 1

Like terms: 7B.

−6 + 7B = 1

Take away −6 on both sides.

7B = 7

Divide by 7: B = 1.

B = 1

And A = 2 minus B, that is 1.

A = 2 − 1 = 1

So aₙ = (−3)ⁿ + 4ⁿ.

a_n = (−3)ⁿ + 4ⁿ

6 Check

Check a₂.

The rule: 1 times 1, plus 12 times 2: 25.

rule: a₂ = 1 × 1 + 12 × 2 = 25

The formula with n = 2 gives 25 too.

Right!

formula: a₂ = 25

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