a_1 = 2, a_n = 3a_(n-1) - 1, find the first 5 terms Step by Step

Step by step

The question

Question: a₁ = 2 and aₙ = 3aₙ₋₁ − 1.

Find the first 5 terms.

a₁ = 2, a_n = 3a_n−1 − 1

aₙ₋₁ means the term just before aₙ.

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

a_n−1 = the term before a_n

Careful!

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

For term 2 we put in a₁, which is 2.

Careful: not a₂ = 3 × 1 − 1 = 2 ✗

1 Start with a₁

The start is given: a₁ = 2.

a₁ = 2

2 Use the rule again and again

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

We put in 2.

3 × 2 − 1 is 5.

a₂ = 3 × a₁ − 1 = 3 × 2 − 1 = 5

Term 3: the term before is a₂, which is 5.

3 × 5 − 1 is 14.

a₃ = 3 × a₂ − 1 = 3 × 5 − 1 = 14

Term 4: 3 × 14 − 1 is 41.

a₄ = 3 × a₃ − 1 = 3 × 14 − 1 = 41

Term 5: 3 × 41 − 1 is 122.

a₅ = 3 × a₄ − 1 = 3 × 41 − 1 = 122

3 Write the list

So the first 5 terms are 2, 5, 14, 41, 122.

= 2, 5, 14, 41, 122

4 Check

Let's check by going back.

The rule did times 3, minus 1.

Going back, we do the opposite: add 1, then divide by 3.

back: + 1, then ÷ 3

From 122: (122 + 1) divided by 3 is 41, term 4.

(122 + 1) ÷ 3 = 41

(41 + 1) ÷ 3 = 14

(14 + 1) ÷ 3 = 5

We land on 2, the start.

It works!

(5 + 1) ÷ 3 = 2

Let's look back.

First: start with a₁.

Then: use the rule again and again.

Then: write the list.

Last: check.

Answer