3, 6, 12, 24, ..., find the 8th term Step by Step

Step by step

The question

Question: 3, 6, 12, 24, …: find term 8.

A sequence is a list of numbers in order.

3, 6, 12, 24, ...

1 Find the ratio

We multiply by 2 every time.

A list like this is a geometric sequence.

The number we multiply by is called the ratio: r = 2.

r = 2

2 Find the rule

Term 1 is 3.

a₁ = 3

a₁ is term 1, a₂ is term 2.

So aₙ is term number n.

a_n = term number n

Term 2: we multiply by 2 one time.

a₂ = 3 × 2

Term 3: we multiply by 2 two times.

That is 2².

a₃ = 3 × 2 × 2 = 3 × 2²

We multiply one time less than the term number.

times we multiply by 2 = term number − 1

So for term n, we multiply by 2, (n − 1) times.

That is the power n − 1.

a_n = 3 × 2ⁿ⁻¹

This rule works for every geometric sequence: the first term, times the ratio to the power (n − 1).

a_n = a₁ × rⁿ⁻¹

3 Find term 8

For term 8, n is 8.

We put 8 in place of n.

a₈ = 3 × 2^{(8 − 1)}

Brackets come first.

Order: ( ) → x² √ → × ÷ → + −

8 minus 1 is 7.

= 3 × 2⁷

Powers come before times.

2⁷ means 2 times 2 times 2 times 2 times 2 times 2 times 2.

2⁷ = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128

That is 128.

= 3 × 128

3 times 128 is 384.

a₈ = 384

So term 8 is 384.

4 Check

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

Put n = 4.

a₄ = 3 × 2⁴⁻¹ = 3 × 2³ = 3 × 8 = 24

We get 24.

That is term 4 in the list.

So the rule is right!

Let's look back.

First: find the ratio.

Then: find the rule.

Then: find term 8.

Last: check.

Answer