Step by step
The question
Question: a₁ = 5 and aₙ = aₙ₋₁ + 6.
Find term 6, a₆.
a1 = 5, an = an−1 + 6
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 5.
Careful: not a2 = 1 + 6 = 7 ✗
1 Start with a₁
The start is given: a₁ = 5.
a1 = 5
2 Use the rule again and again
Term 2: n = 2, so the term before is a₁.
We put in 5.
5 + 6 is 11.
a2 = a1 + 6 = 5 + 6 = 11
Term 3: the term before is a₂, which is 11.
11 + 6 is 17.
a3 = a2 + 6 = 11 + 6 = 17
Term 4: 17 + 6 is 23.
a4 = a3 + 6 = 17 + 6 = 23
Term 5: 23 + 6 is 29.
a5 = a4 + 6 = 23 + 6 = 29
Term 6: 29 + 6 is 35.
a6 = a5 + 6 = 29 + 6 = 35
So a₆ = 35.
3 Check
Let's check by going back.
The rule did plus 6.
Going back, we do the opposite: take away 6.
back: − 6
From 35: 35 − 6 is 29, term 5.
35 − 6 = 29
29 − 6 = 23
23 − 6 = 17
17 − 6 = 11
We land on 5, the start.
It works!
11 − 6 = 5
Let's look back.
First: start with a₁.
Then: use the rule again and again.
Last: check.
Answer
a6 = a5 + 6 = 29 + 6 = 35