Calculate 1.2, 1.5, 1.8, ..., nth term Step by Step

Step by step

The question

Question: 1.2, 1.5, 1.8, …: find the rule for term n.

A sequence is a list of numbers in order.

1.2, 1.5, 1.8, ...

1 Find the difference

We add 0.3 every time.

A list like this is an arithmetic sequence.

The number we add is called the difference: d = 0.3.

d = 0.3

2 Find the rule

Term 1 is 1.2.

a₁ = 1.2

a₁ is term 1, a₂ is term 2.

So aₙ is term number n.

a_n = term number n

Term 2: we add 0.3 one time.

a₂ = 1.2 + 1 × 0.3

Term 3: we add it two times.

a₃ = 1.2 + 2 × 0.3

We add 0.3 one time less than the term number.

times we add 0.3 = term number − 1

So for term n, we add 0.3, (n − 1) times.

a_n = 1.2 + (n − 1) × 0.3

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 0.3.

Every part inside is multiplied: n times 0.3 is 0.3n.

−1 times 0.3 is −0.3.

a_n = 1.2 + 0.3n − 0.3

Plain numbers go together: 1.2 − 0.3 is 0.9.

1.2 − 0.3 = 0.9

So the rule is aₙ = 0.3n + 0.9.

a_n = 0.3n + 0.9

4 Check

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

Put n = 3.

a₃ = 310(3) + 910 = 0.9 + 0.9 = 1.8

We get 1.8.

That is term 3 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