Step by step
The question
Question: a₁ = 10 and aₙ = aₙ₋₁ − 3.
Find the first 5 terms.
a1 = 10, an = an−1 − 3
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 10.
Careful: not a2 = 1 − 3 = −2 ✗
1 Start with a₁
The start is given: a₁ = 10.
a1 = 10
2 Use the rule again and again
Term 2: n = 2, so the term before is a₁.
We put in 10.
10 − 3 is 7.
a2 = a1 − 3 = 10 − 3 = 7
Term 3: the term before is a₂, which is 7.
7 − 3 is 4.
a3 = a2 − 3 = 7 − 3 = 4
Term 4: 4 − 3 is 1.
a4 = a3 − 3 = 4 − 3 = 1
Term 5: 1 − 3 is −2.
a5 = a4 − 3 = 1 − 3 = −2
3 Write the list
So the first 5 terms are 10, 7, 4, 1, −2.
= 10, 7, 4, 1, −2
4 Check
Let's check by going back.
The rule did minus 3.
Going back, we do the opposite: add 3.
back: + 3
From −2: −2 + 3 is 1, term 4.
−2 + 3 = 1
1 + 3 = 4
4 + 3 = 7
We land on 10, the start.
It works!
7 + 3 = 10
Let's look back.
First: start with a₁.
Then: use the rule again and again.
Then: write the list.
Last: check.
Answer
= 10, 7, 4, 1, −2