Step by step
The question
Question: 1, −2, 4, −8, …: find term 9.
A sequence is a list of numbers in order.
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.
The minus makes the sign change every time.
2 Find the rule
Term 1 is 1.
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
Term 2: we multiply by −2 one time.
Term 3: we multiply by −2 two times.
That is (−2)².
We multiply one time less than the term number.
So for term n, we multiply by −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 9
For term 9, n is 9.
We put 9 in place of n.
Brackets come first.
9 minus 1 is 8.
Powers come before times.
(−2)⁸ means −2 times −2 times −2 times −2 times −2 times −2 times −2 times −2.
Minus times minus is plus.
That is 256.
1 times 256 is 256.
So term 9 is 256.
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 9.
Last: check.