Solve a_n = 7a_(n-1) - 10a_(n-2), a_0 = 3, a

Step by step

The question

Question: solve aₙ = 7aₙ₋₁ − 10aₙ₋₂, with a₀ = 3 and a₁ = 9.

a_n = 7a_{n − 1} − 10a_{n − 2}, a₀ = 3, a₁ = 9

1 Everything on one side

Move everything to one side.

Each term changes its sign when it moves.

a_n − 7a_{n − 1} + 10a_{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 − 7x^{n − 1} + 10x^{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 − 7x + 10 = 0

÷ x^{n − 2} on both sides

Careful!

Watch the signs.

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

Careful: not x² − 7x − 10 = 0 ✗

3 Solve it

Factor: we need two numbers that multiply to 10 and add to −7: −2 and −5.

(x − 2)(x − 5) = 0

(−2) × (−5) = 10, (−2) + (−5) = −7

So x = 2 or x = 5.

x = 2 or 5

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 5 to the n.

a_n = A × 2ⁿ + B × 5ⁿ

5 Use the start values

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

So A + B = 3.

A + B = 3 ①

n = 1: 2A + 5B = 9.

2A + 5B = 9 ②

From equation 1: A = 3 − B.

A = 3 − B

Put it into equation 2.

2(3 − B) + 5B = 9

Open the bracket.

6 − 2B + 5B = 9

Like terms: 3B.

6 + 3B = 9

Take away 6 on both sides.

3B = 3

Divide by 3: B = 1.

B = 1

And A = 3 minus B, that is 2.

A = 3 − 1 = 2

So aₙ = 2 × 2ⁿ + 5ⁿ.

a_n = 2 × 2ⁿ + 5ⁿ

6 Check

Check a₂.

The rule: 7 times 9, minus 10 times 3: 33.

rule: a₂ = 7 × 9 − 10 × 3 = 33

The formula with n = 2 gives 33 too.

Right!

formula: a₂ = 33

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