Solve a_n = a_(n-1) + 6a_(n-2), a_0 = 0, a

Step by step

The question

Question: solve aₙ = aₙ₋₁ + 6aₙ₋₂, with a₀ = 0 and a₁ = 5.

a_n = a_{n − 1} + 6a_{n − 2}, a₀ = 0, a₁ = 5

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: 2 and −3.

(x + 2)(x − 3) = 0

2 × (−3) = −6, 2 + (−3) = −1

So x = −2 or x = 3.

x = −2 or 3

4 The general form

Each root gives a power that works.

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

a_n = A × (−2)ⁿ + B × 3ⁿ

5 Use the start values

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

So A + B = 0.

A + B = 0 ①

n = 1: −2A + 3B = 5.

-2A + 3B = 5 ②

From equation 1: A = 0 − B.

A = 0 − B

Put it into equation 2.

-2(0 − B) + 3B = 5

Open the bracket.

0 + 2B + 3B = 5

Like terms: 5B.

0 + 5B = 5

Take away 0 on both sides.

5B = 5

Divide by 5: B = 1.

B = 1

And A = 0 minus B, that is −1.

A = 0 − 1 = −1

So aₙ = −(−2)ⁿ + 3ⁿ.

a_n = −(−2)ⁿ + 3ⁿ

6 Check

Check a₂.

The rule: 1 times 5, plus 6 times 0: 5.

rule: a₂ = 1 × 5 + 6 × 0 = 5

The formula with n = 2 gives 5 too.

Right!

formula: a₂ = 5

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