Step by step
The question
Question: 50, 46, 42, 38, …: find term 20.
A sequence is a list of numbers in order.
1 Find the difference
From 50 to 46, we take away 4.
From 46 to 42, we take away 4 again.
We take away 4 every time.
A list like this is an arithmetic sequence.
Taking away 4 is the same as adding −4.
The number we add is called the difference: d = −4.
2 Find the rule
Term 1 is 50.
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 Find term 20
For term 20, n is 20.
We put 20 in place of n.
Everyone uses the same order, so we all get the same answer.
Brackets come first.
20 minus 1 is 19.
Times comes before plus.
19 has no sign, so it is plus.
Plus times minus is minus.
19 times 4 is 76.
So we get −76.
Adding a negative is the same as taking away.
So 50 + (−76) becomes 50 − 76.
76 is more than 50.
We take away more than we have.
50 of the 76 takes us down to 0.
Then 76 − 50 = 26 more, below 0.
We land on −26.
So term 20 is −26.
4 Check
Let's check the rule with a term we know.
Put n = 4.
We get 38.
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: find term 20.
Last: check.