Solve a_n = 2a_(n-1) - 3, a_0 = 5

Step by step

The question

Question: solve aₙ = 2aₙ₋₁ − 3, starting with a₀ = 5.

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 = 5.

Then 2 times 5, minus 3.

That is 7.

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 2c − 3, and it must be c again.

Careful!

c is on both sides.

We solve for it like any equation.

Take away 2c on both sides.

c minus 2c is −c.

Divide by −1: c = 3.

Check it: 2 times 3, minus 3 is 3 again.

3 The distance from it

Now the distance from c: each term minus 3.

Look at the new column: 2, 4, 8, 16, 32.

Each one is 2 times the one before.

The distance from c grows by times 2.

4 The formula

The distance starts at 2 and is multiplied by 2 each step: 2 times 2 to the n.

Add 3 on both sides.

2 is one more 2, so the powers join: 2 times 2 to the n is 2 to the n + 1.

5 Check

Check n = 3 with the formula: it gives 19.

The table says 19 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.