Step by step
The question
Question: 2, 6, 18, 54, …: find term 6.
A sequence is a list of numbers in order.
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.
2 Find the rule
Term 1 is 2.
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
Term 2: we multiply by 3 one time.
Term 3: we multiply by 3 two times.
That is 3².
We multiply one time less than the term number.
So for term n, we multiply by 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).
3 Find term 6
For term 6, n is 6.
We put 6 in place of n.
Brackets come first.
6 minus 1 is 5.
Powers come before times.
3⁵ means 3 times 3 times 3 times 3 times 3.
That is 243.
2 times 243 is 486.
So term 6 is 486.
4 Check
Let's check the rule with a term we know.
Put n = 4.
We get 54.
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 6.
Last: check.