Step by step
The question
Question: 4, 9, 14, 19, …: find the rule for term n.
A sequence is a list of numbers in order.
1 Find the difference
From 4 to 9, we add 5.
From 9 to 14, we add 5 again.
We add 5 every time.
A list like this is an arithmetic sequence.
The number we add is called the difference: d = 5.
2 Find the rule
Term 1 is 4.
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
Term 2: we add 5 one time.
Term 3: we add it two times.
We add 5 one time less than the term number.
So for term n, we add 5, (n − 1) times.
This rule works for every arithmetic sequence: the first term, plus (n − 1) times the difference.
3 Make the rule simpler
Open the bracket: each part inside is times 5.
n times 5 is 5n.
The − 1 inside, times 5, is −5.
Plain numbers go together: 4 − 5 is −1.
So the rule is aₙ = 5n − 1.
4 Check
Let's check the rule with a term we know.
Put n = 4.
We get 19.
That is term 4 in the list.
So the rule is right!
Let's look back.
First: find the difference.
Then: find the rule.
Then: make the rule simpler.
Last: check.