Step by step
The question
Question: a₁ = 2 and aₙ = 3aₙ₋₁.
Find the first 5 terms.
a1 = 2, an = 3an−1
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 2.
Careful: not a2 = 3 × 1 = 3 ✗
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 is 6.
a2 = 3 × a1 = 3 × 2 = 6
Term 3: the term before is a₂, which is 6.
3 × 6 is 18.
a3 = 3 × a2 = 3 × 6 = 18
Term 4: 3 × 18 is 54.
a4 = 3 × a3 = 3 × 18 = 54
Term 5: 3 × 54 is 162.
a5 = 3 × a4 = 3 × 54 = 162
3 Write the list
So the first 5 terms are 2, 6, 18, 54, 162.
= 2, 6, 18, 54, 162
4 Check
Let's check by going back.
The rule did times 3.
Going back, we do the opposite: divide by 3.
back: ÷ 3
From 162: 162 divided by 3 is 54, term 4.
162 ÷ 3 = 54
54 ÷ 3 = 18
18 ÷ 3 = 6
We land on 2, the start.
It works!
6 ÷ 3 = 2
Let's look back.
First: start with a₁.
Then: use the rule again and again.
Then: write the list.
Last: check.
Answer
= 2, 6, 18, 54, 162