Step by step
The question
Question: 7, 11, 15, 19, …: find the rule for term n.
A sequence is a list of numbers in order.
1 Find the difference
From 7 to 11, we add 4.
From 11 to 15, we add 4 again.
We add 4 every time.
A list like this is an arithmetic sequence.
The number we add is called the difference: d = 4.
2 Find the rule
Term 1 is 7.
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
Term 2: we add 4 one time.
Term 3: we add it two times.
We add 4 one time less than the term number.
So for term n, we add 4, (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 4.
n times 4 is 4n.
The − 1 inside, times 4, is −4.
Plain numbers go together: 7 − 4 is 3.
So the rule is aₙ = 4n + 3.
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.