Step by step
The question
Question: 64, 32, 16, 8, …: find term 6.
A sequence is a list of numbers in order.
1 Find the ratio
We multiply by 1/2 every time.
A list like this is a geometric sequence.
The number we multiply by is called the ratio: r = 1/2.
Times 1/2 is the same as dividing by 2.
2 Find the rule
Term 1 is 64.
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
Term 2: we multiply by 1/2 one time.
Term 3: we multiply by 1/2 two times.
That is (1/2)².
We multiply one time less than the term number.
So for term n, we multiply by 1/2, (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.
(1/2)⁵ means 1/2 times 1/2 times 1/2 times 1/2 times 1/2.
That is 1/32.
Before we multiply, we make the numbers smaller.
64 on top and 32 at the bottom both divide by 32.
A top and a bottom divided by the same number: the answer stays the same.
So we multiply smaller numbers.
2 times 1 is 2.
So term 6 is 2.
4 Check
Let's check the rule with a term we know.
Put n = 4.
We get 8.
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.