Step by step
The question
Question: 20, 16, 12, 8, …: find the rule for term n.
A sequence is a list of numbers in order.
1 Find the difference
We take away 4 every time.
A list like this is an arithmetic sequence.
The number we add is called the difference: d = −4.
It is negative, because we go down.
2 Find the rule
Term 1 is 20.
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
The bracket is times (−4).
Every part inside is multiplied: n times (−4) is −4n.
−1 times (−4) is 4.
Plain numbers go together: 20 + 4 is 24.
So the rule is aₙ = −4n + 24.
4 Check
Let's check the rule with a term we know.
Put n = 4.
We get 8.
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.