Step by step
The question
Question: a₁ = 1 and aₙ = 4aₙ₋₁.
Find the first 4 terms.
a1 = 1, an = 4an−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 1.
Careful: not a2 = 4 × 1 = 4 ✗
1 Start with a₁
The start is given: a₁ = 1.
a1 = 1
2 Use the rule again and again
Term 2: n = 2, so the term before is a₁.
We put in 1.
4 × 1 is 4.
a2 = 4 × a1 = 4 × 1 = 4
Term 3: the term before is a₂, which is 4.
4 × 4 is 16.
a3 = 4 × a2 = 4 × 4 = 16
Term 4: 4 × 16 is 64.
a4 = 4 × a3 = 4 × 16 = 64
3 Write the list
So the first 4 terms are 1, 4, 16, 64.
= 1, 4, 16, 64
4 Check
Let's check by going back.
The rule did times 4.
Going back, we do the opposite: divide by 4.
back: ÷ 4
From 64: 64 divided by 4 is 16, term 3.
64 ÷ 4 = 16
16 ÷ 4 = 4
We land on 1, the start.
It works!
4 ÷ 4 = 1
Let's look back.
First: start with a₁.
Then: use the rule again and again.
Then: write the list.
Last: check.
Answer
= 1, 4, 16, 64