Step by step
The question
Question: a₁ = 4 and aₙ = 2aₙ₋₁.
Find term 5, a₅.
a1 = 4, an = 2an−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 4.
Careful: not a2 = 2 × 1 = 2 ✗
1 Start with a₁
The start is given: a₁ = 4.
a1 = 4
2 Use the rule again and again
Term 2: n = 2, so the term before is a₁.
We put in 4.
2 × 4 is 8.
a2 = 2 × a1 = 2 × 4 = 8
Term 3: the term before is a₂, which is 8.
2 × 8 is 16.
a3 = 2 × a2 = 2 × 8 = 16
Term 4: 2 × 16 is 32.
a4 = 2 × a3 = 2 × 16 = 32
Term 5: 2 × 32 is 64.
a5 = 2 × a4 = 2 × 32 = 64
So a₅ = 64.
3 Check
Let's check by going back.
The rule did times 2.
Going back, we do the opposite: divide by 2.
back: ÷ 2
From 64: 64 divided by 2 is 32, term 4.
64 ÷ 2 = 32
32 ÷ 2 = 16
16 ÷ 2 = 8
We land on 4, the start.
It works!
8 ÷ 2 = 4
Let's look back.
First: start with a₁.
Then: use the rule again and again.
Last: check.
Answer
a5 = 2 × a4 = 2 × 32 = 64