a_1 = 1, a_n = 2a_(n-1) + 1, find the first 5 terms

Step by step

The question

Question: a₁ = 1 and aₙ = 2aₙ₋₁ + 1.

Find the first 5 terms.

aₙ₋₁ means the term just before aₙ.

A rule that makes each term from the term before is called recursive.

Careful!

aₙ₋₁ is the term before, not the number n − 1.

For term 2 we put in a₁, which is 1.

1 Start with a₁

The start is given: a₁ = 1.

2 Use the rule again and again

Term 2: n = 2, so the term before is a₁.

We put in 1.

2 × 1 + 1 is 3.

Term 3: the term before is a₂, which is 3.

2 × 3 + 1 is 7.

Term 4: 2 × 7 + 1 is 15.

Term 5: 2 × 15 + 1 is 31.

3 Write the list

So the first 5 terms are 1, 3, 7, 15, 31.

4 Check

Let's check by going back.

The rule did times 2, plus 1.

Going back, we do the opposite: take away 1, then divide by 2.

From 31: (31 − 1) divided by 2 is 15, term 4.

We land on 1, the start.

It works!

Let's look back.

First: start with a₁.

Then: use the rule again and again.

Then: write the list.

Last: check.