Calculate 6, 11, 16, 21, ..., nth term Step by Step

Step by step

The question

Question: 6, 11, 16, 21, …: find the rule for term n.

A sequence is a list of numbers in order.

6, 11, 16, 21, ...

1 Find the difference

We add 5 every time.

A list like this is an arithmetic sequence.

The number we add is called the difference: d = 5.

d = 5

2 Find the rule

Term 1 is 6.

a₁ = 6

a₁ is term 1, a₂ is term 2.

So aₙ is term number n.

a_n = term number n

Term 2: we add 5 one time.

a₂ = 6 + 1 × 5

Term 3: we add it two times.

a₃ = 6 + 2 × 5

We add 5 one time less than the term number.

times we add 5 = term number − 1

So for term n, we add 5, (n − 1) times.

a_n = 6 + (n − 1) × 5

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 5.

Every part inside is multiplied: n times 5 is 5n.

−1 times 5 is −5.

a_n = 6 + 5n − 5

Plain numbers go together: 6 − 5 is 1.

6 − 5 = 1

So the rule is aₙ = 5n + 1.

a_n = 5n + 1

4 Check

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

Put n = 4.

a₄ = 5(4) + 1 = 20 + 1 = 21

We get 21.

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