Step by step
The question
Question: a₁ = 0 and aₙ = 3aₙ₋₁ + 2.
Find term 5, a₅.
a1 = 0, an = 3an−1 + 2
aₙ₋₁ means the term just before aₙ.
A rule that makes each term from the term before is called recursive.
an−1 = the term before an
Careful!
aₙ₋₁ is the term before, not the number n − 1.
For term 2 we put in a₁, which is 0.
Careful: not a2 = 3 × 1 + 2 = 5 ✗
1 Start with a₁
The start is given: a₁ = 0.
a1 = 0
2 Use the rule again and again
Term 2: n = 2, so the term before is a₁.
We put in 0.
3 × 0 + 2 is 2.
a2 = 3 × a1 + 2 = 3 × 0 + 2 = 2
Term 3: the term before is a₂, which is 2.
3 × 2 + 2 is 8.
a3 = 3 × a2 + 2 = 3 × 2 + 2 = 8
Term 4: 3 × 8 + 2 is 26.
a4 = 3 × a3 + 2 = 3 × 8 + 2 = 26
Term 5: 3 × 26 + 2 is 80.
a5 = 3 × a4 + 2 = 3 × 26 + 2 = 80
So a₅ = 80.
3 Check
Let's check by going back.
The rule did times 3, plus 2.
Going back, we do the opposite: take away 2, then divide by 3.
back: − 2, then ÷ 3
From 80: (80 − 2) divided by 3 is 26, term 4.
(80 − 2) ÷ 3 = 26
(26 − 2) ÷ 3 = 8
(8 − 2) ÷ 3 = 2
We land on 0, the start.
It works!
(2 − 2) ÷ 3 = 0
Let's look back.
First: start with a₁.
Then: use the rule again and again.
Last: check.
Answer
a5 = 3 × a4 + 2 = 3 × 26 + 2 = 80