Step by step
The question
Question: 81, 27, 9, …: find the rule for term n.
A sequence is a list of numbers in order.
1 Find the ratio
We multiply by 1/3 every time.
A list like this is a geometric sequence.
The number we multiply by is called the ratio: r = 1/3.
Times 1/3 is the same as dividing by 3.
2 Find the rule
Term 1 is 81.
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
Term 2: we multiply by 1/3 one time.
Term 3: we multiply by 1/3 two times.
That is (1/3)².
We multiply one time less than the term number.
So for term n, we multiply by 1/3, (n − 1) times.
That is the power n − 1.
This rule works for every geometric sequence: the first term, times the ratio to the power (n − 1).
So the rule is aₙ = 81 × (1/3)ⁿ⁻¹.
3 Check
Let's check the rule with a term we know.
Put n = 3.
We get 9.
That is term 3 in the list.
So the rule is right!
Let's look back.
First: find the ratio.
Then: find the rule.
Last: check.