10, 7, 4, 1, ..., find the 12th term

Step by step

The question

Question: 10, 7, 4, 1, …: find term 12.

A sequence is a list of numbers in order.

1 Find the difference

We take away 3 every time.

A list like this is an arithmetic sequence.

The number we add is called the difference: d = −3.

It is negative, because we go down.

2 Find the rule

Term 1 is 10.

a₁ is term 1, a₂ is term 2.

So aₙ is term number n.

Term 2: we add (−3) one time.

Term 3: we add it two times.

We add (−3) one time less than the term number.

So for term n, we add (−3), (n − 1) times.

This rule works for every arithmetic sequence: the first term, plus (n − 1) times the difference.

3 Find term 12

For term 12, n is 12.

We put 12 in place of n.

Brackets come first.

12 minus 1 is 11.

Times comes before plus.

Plus times minus is minus.

11 times 3 is 33.

So we get −33.

Adding a negative is the same as taking away.

So 10 + (−33) becomes 10 − 33.

We take away more than we have.

Start at 10.

Go down 33.

We pass 0 and land on −23.

So term 12 is −23.

4 Check

Let's check the rule with a term we know.

Put n = 4.

We get 1.

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 12.

Last: check.