Solve a_n = 8a_(n-1) - 15a_(n-2), a_0 = 2, a

Step by step

The question

Question: solve aₙ = 8aₙ₋₁ − 15aₙ₋₂, with a₀ = 2 and a₁ = 8.

a_n = 8a_{n − 1} − 15a_{n − 2}, a₀ = 2, a₁ = 8

1 Everything on one side

Move everything to one side.

Each term changes its sign when it moves.

a_n − 8a_{n − 1} + 15a_{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 − 8x^{n − 1} + 15x^{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 − 8x + 15 = 0

÷ x^{n − 2} on both sides

Careful!

Watch the signs.

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

Careful: not x² − 8x − 15 = 0 ✗

3 Solve it

Factor: we need two numbers that multiply to 15 and add to −8: −3 and −5.

(x − 3)(x − 5) = 0

(−3) × (−5) = 15, (−3) + (−5) = −8

So x = 3 or x = 5.

x = 3 or 5

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 5 to the n.

a_n = A × 3ⁿ + B × 5ⁿ

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 + 5B = 8.

3A + 5B = 8 ②

From equation 1: A = 2 − B.

A = 2 − B

Put it into equation 2.

3(2 − B) + 5B = 8

Open the bracket.

6 − 3B + 5B = 8

Like terms: 2B.

6 + 2B = 8

Take away 6 on both sides.

2B = 2

Divide by 2: B = 1.

B = 1

And A = 2 minus B, that is 1.

A = 2 − 1 = 1

So aₙ = 3ⁿ + 5ⁿ.

a_n = 3ⁿ + 5ⁿ

6 Check

Check a₂.

The rule: 8 times 8, minus 15 times 2: 34.

rule: a₂ = 8 × 8 − 15 × 2 = 34

The formula with n = 2 gives 34 too.

Right!

formula: a₂ = 34

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