Step by step
The question
Question: a geometric sequence has a₁ = 2 and a₃ = −18.
Find the rule for term n.
1 Count the jumps
Term 1 is 2.
Term 3 is −18.
We don't know the terms between.
In a geometric sequence we multiply by the same number r every time.
How many jumps?
3 take away 1: 2 jumps.
2 Find r
Start at 2, multiply by r 2 times, and we get −18.
r times itself 2 times is r².
To remove times 2, we divide by 2.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
2 divided by 2 is 1.
So just r² is left.
−18 divided by 2 is −9.
Which number times itself is −9?
A plus times a plus is plus.
A minus times a minus is plus too.
So no real number works.
So there is no real r.
There is no answer: no geometric sequence has these two terms.
Let's look back.
First: count the jumps.
Last: find r.