-10, -6, -2, 2, ..., find the 25th term Step by Step

Step by step

The question

Question: −10, −6, −2, 2, …: find term 25.

A sequence is a list of numbers in order.

−10, −6, −2, 2, ...

1 Find the difference

We add 4 every time.

A list like this is an arithmetic sequence.

The number we add is called the difference: d = 4.

d = 4

2 Find the rule

Term 1 is −10.

a₁ = −10

a₁ is term 1, a₂ is term 2.

So aₙ is term number n.

a_n = term number n

Term 2: we add 4 one time.

a₂ = −10 + 1 × 4

Term 3: we add it two times.

a₃ = −10 + 2 × 4

We add 4 one time less than the term number.

times we add 4 = term number − 1

So for term n, we add 4, (n − 1) times.

a_n = −10 + (n − 1) × 4

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 25

For term 25, n is 25.

We put 25 in place of n.

a₂₅ = −10 + (25 − 1) × 4

Brackets come first.

Order: ( ) → x² √ → × ÷ → + −

25 minus 1 is 24.

= −10 + 24 × 4

Times comes before plus.

24 times 4 is 96.

= −10 + 96

Start at −10.

Go up 96.

We land on 86.

a₂₅ = 86

So term 25 is 86.

4 Check

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

Put n = 4.

a₄ = −10 + (4 − 1) × 4 = −10 + 3 × 4

= −10 + 12 = 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 25.

Last: check.

Answer