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

Step by step

The question

Question: a₁ = −1 and aₙ = 2aₙ₋₁ + 5.

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) + 5 is 3.

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

2 × 3 + 5 is 11.

Term 4: 2 × 11 + 5 is 27.

Term 5: 2 × 27 + 5 is 59.

3 Write the list

So the first 5 terms are −1, 3, 11, 27, 59.

4 Check

Let's check by going back.

The rule did times 2, plus 5.

Going back, we do the opposite: take away 5, then divide by 2.

From 59: (59 − 5) divided by 2 is 27, 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.