0.5, 1, 1.5, 2, ..., find the 10th term Step by Step

Step by step

The question

Question: 0.5, 1, 1.5, 2, …: find term 10.

A sequence is a list of numbers in order.

0.5, 1, 1.5, 2, ...

1 Find the difference

We add 0.5 every time.

A list like this is an arithmetic sequence.

The number we add is called the difference: d = 0.5.

d = 0.5

2 Find the rule

Term 1 is 0.5.

a₁ = 0.5

a₁ is term 1, a₂ is term 2.

So aₙ is term number n.

a_n = term number n

Term 2: we add 0.5 one time.

a₂ = 0.5 + 1 × 0.5

Term 3: we add it two times.

a₃ = 0.5 + 2 × 0.5

We add 0.5 one time less than the term number.

times we add 0.5 = term number − 1

So for term n, we add 0.5, (n − 1) times.

a_n = 0.5 + (n − 1) × 0.5

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

a_n = a₁ + (n − 1) × d

3 Find term 10

For term 10, n is 10.

We put 10 in place of n.

a₁₀ = 0.5 + (10 − 1) × 0.5

Brackets come first.

Order: ( ) → x² √ → × ÷ → + −

10 minus 1 is 9.

= 0.5 + 9 × 0.5

Times comes before plus.

9 × 5 = 45

1 digit after the point → 4.5

Forget the points: 9 times 5 is 45.

The numbers have 1 digit after the point.

So the answer has 1 too: 4.5.

= 0.5 + 4.5

Line up the decimal points.

0.5 plus 4.5 is 5.

a₁₀ = 5

So term 10 is 5.

4 Check

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

Put n = 4.

a₄ = 0.5 + (4 − 1) × 0.5 = 0.5 + 3 × 0.5

= 0.5 + 1.5 = 2

We get 2.

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 10.

Last: check.

Answer