Calculate 1, 4, 7, 10, ..., nth term Step by Step

Step by step

The question

Question: 1, 4, 7, 10, …: find the rule for term n.

A sequence is a list of numbers in order.

1, 4, 7, 10, ...

1 Find the difference

We add 3 every time.

A list like this is an arithmetic sequence.

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

d = 3

2 Find the rule

Term 1 is 1.

a₁ = 1

a₁ is term 1, a₂ is term 2.

So aₙ is term number n.

a_n = term number n

Term 2: we add 3 one time.

a₂ = 1 + 1 × 3

Term 3: we add it two times.

a₃ = 1 + 2 × 3

We add 3 one time less than the term number.

times we add 3 = term number − 1

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

a_n = 1 + (n − 1) × 3

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

a_n = a₁ + (n − 1) × d

3 Make the rule simpler

The bracket is times 3.

Every part inside is multiplied: n times 3 is 3n.

−1 times 3 is −3.

a_n = 1 + 3n − 3

Plain numbers go together: 1 − 3 is −2.

1 − 3 = −2

So the rule is aₙ = 3n − 2.

a_n = 3n − 2

4 Check

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

Put n = 4.

a₄ = 3(4) − 2 = 12 − 2 = 10

We get 10.

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.

Answer