Step by step
The question
Question: a geometric sequence has a₂ = 12 and a₅ = −96.
Find the rule for term n.
1 Count the jumps
Term 2 is 12.
Term 5 is −96.
We don't know the terms between.
In a geometric sequence we multiply by the same number r every time.
How many jumps?
5 take away 2: 3 jumps.
Careful!
We count the jumps, not the terms.
4 terms have only 3 jumps between them.
2 Find r
Start at 12, multiply by r 3 times, and we get −96.
r times itself 3 times is r³.
To remove times 12, we divide by 12.
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.
12 divided by 12 is 1.
So just r³ is left.
−96 divided by 12 is −8.
Which number times itself 3 times is −8?
−2.
A minus, an odd number of times, stays minus.
So r = −2.
3 Find the first term
a₁ is 1 jump before a₂.
Going back, we divide by r each time.
12 divided by −2 is −6.
4 Write the rule
The rule from the sequences lesson: the first term, times the ratio to the power (n − 1).
Put in a₁ = −6 and r = −2.
So the rule is aₙ = −6 × (−2)ⁿ⁻¹.
5 Check
Let's check with the terms we were given.
Put n = 2: the power is 1.
We get 12.
And n = 5 gives −96.
Both work!
Let's look back.
First: count the jumps.
Then: find r.
Then: find the first term.
Then: write the rule.
Last: check.