Step by step
The question
Question: a geometric sequence has a₂ = −4 and a₃ = 8.
Find the rule for term n.
1 Count the jumps
Term 2 is −4.
Term 3 is 8.
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 2: 1 jump.
2 Find r
Start at −4, multiply by r 1 times, and we get 8.
To remove times −4, we divide by −4.
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.
−4 divided by −4 is 1.
So just r¹ is left.
8 divided by −4 is −2.
So r = −2.
3 Find the first term
a₁ is 1 jump before a₂.
Going back, we divide by r each time.
−4 divided by −2 is 2.
4 Write the rule
The rule from the sequences lesson: the first term, times the ratio to the power (n − 1).
Put in a₁ = 2 and r = −2.
So the rule is aₙ = 2 × (−2)ⁿ⁻¹.
5 Check
Let's check with the terms we were given.
Put n = 2: the power is 1.
We get −4.
And n = 3 gives 8.
Both work!
Let's look back.
First: count the jumps.
Then: find r.
Then: find the first term.
Then: write the rule.
Last: check.