Solve a_n = 4a_(n-1) + 3, a_0 = 0

Step by step

The question

Question: solve aₙ = 4aₙ₋₁ + 3, starting with a₀ = 0.

Solve means: find a formula for aₙ, so we can get any term without the ones before.

1 First terms

First, a few terms.

Each one comes from the one before.

We start with a₀ = 0.

Then 4 times 0, plus 3.

That is 3.

And so on.

2 The number that stays the same

Is there a number that the rule keeps the same?

Call it c.

If a term were c, the next would be 4c + 3, and it must be c again.

Careful!

c is on both sides.

We solve for it like any equation.

Take away 4c on both sides.

c minus 4c is −3c.

Divide by −3: c = −1.

Check it: 4 times −1, plus 3 is −1 again.

3 The distance from it

Now the distance from c: each term plus 1.

Look at the new column: 1, 4, 16, 64, 256.

Each one is 4 times the one before.

The distance from c grows by times 4.

4 The formula

The distance starts at 1 and is multiplied by 4 each step: 1 times 4 to the n.

Take away 1 on both sides.

5 Check

Check n = 3 with the formula: it gives 63.

The table says 63 too.

Right!

Let's look back.

First: first terms.

Then: the number that stays the same.

Then: the distance from it.

Then: the formula.

Last: check.