Step by step
The question
Question: a₁ = 64 and aₙ = aₙ₋₁ − 9.
Find the first 4 terms.
a1 = 64, an = an−1 − 9
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 64.
Careful: not a2 = 1 − 9 = −8 ✗
1 Start with a₁
The start is given: a₁ = 64.
a1 = 64
2 Use the rule again and again
Term 2: n = 2, so the term before is a₁.
We put in 64.
64 − 9 is 55.
a2 = a1 − 9 = 64 − 9 = 55
Term 3: the term before is a₂, which is 55.
55 − 9 is 46.
a3 = a2 − 9 = 55 − 9 = 46
Term 4: 46 − 9 is 37.
a4 = a3 − 9 = 46 − 9 = 37
3 Write the list
So the first 4 terms are 64, 55, 46, 37.
= 64, 55, 46, 37
4 Check
Let's check by going back.
The rule did minus 9.
Going back, we do the opposite: add 9.
back: + 9
From 37: 37 + 9 is 46, term 3.
37 + 9 = 46
46 + 9 = 55
We land on 64, the start.
It works!
55 + 9 = 64
Let's look back.
First: start with a₁.
Then: use the rule again and again.
Then: write the list.
Last: check.
Answer
= 64, 55, 46, 37