1, 3, 9, 27, ..., find the 7th term Step by Step

Step by step

The question

Question: 1, 3, 9, 27, …: find term 7.

A sequence is a list of numbers in order.

1, 3, 9, 27, ...

1 Find the ratio

We multiply by 3 every time.

A list like this is a geometric sequence.

The number we multiply by is called the ratio: r = 3.

r = 3

2 Find the rule

Term 1 is 1.

a₁ = 1

a₁ is term 1, a₂ is term 2.

So aₙ is term number n.

a_n = term number n

Term 2: we multiply by 3 one time.

a₂ = 1 × 3

Term 3: we multiply by 3 two times.

That is 3².

a₃ = 1 × 3 × 3 = 1 × 3²

We multiply one time less than the term number.

times we multiply by 3 = term number − 1

So for term n, we multiply by 3, (n − 1) times.

That is the power n − 1.

a_n = 1 × 3ⁿ⁻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 7

For term 7, n is 7.

We put 7 in place of n.

a₇ = 1 × 3^{(7 − 1)}

Brackets come first.

Order: ( ) → x² √ → × ÷ → + −

7 minus 1 is 6.

= 1 × 3⁶

Powers come before times.

3⁶ means 3 times 3 times 3 times 3 times 3 times 3.

3⁶ = 3 × 3 × 3 × 3 × 3 × 3 = 729

That is 729.

= 1 × 729

1 times 729 is 729.

a₇ = 729

So term 7 is 729.

4 Check

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

Put n = 4.

a₄ = 1 × 3⁴⁻¹ = 1 × 3³ = 1 × 27 = 27

We get 27.

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 7.

Last: check.

Answer