Step by step
The question
Question: 2, 9, 16, 23, …: find term 40.
A sequence is a list of numbers in order.
1 Find the difference
From 2 to 9, we add 7.
From 9 to 16, we add 7 again.
We add 7 every time.
A list like this is an arithmetic sequence.
The number we add is called the difference: d = 7.
2 Find the rule
Term 1 is 2.
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
Term 2: we add 7 one time.
Term 3: we add it two times.
We add 7 one time less than the term number.
So for term n, we add 7, (n − 1) times.
This rule works for every arithmetic sequence: the first term, plus (n − 1) times the difference.
3 Find term 40
For term 40, n is 40.
We put 40 in place of n.
Everyone uses the same order, so we all get the same answer.
Brackets come first.
40 minus 1 is 39.
Times comes before plus.
39 times 7 is 273.
We write them in a column: ones under ones, tens under tens.
We add each column, from the right.
2 + 3 = 5.
Only 7 in this column, so it comes down: 7.
Only 2 in this column, so it comes down: 2.
2 plus 273 is 275.
So term 40 is 275.
4 Check
Let's check the rule with a term we know.
Put n = 4.
We get 23.
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 40.
Last: check.