Step by step
The question
Question: a geometric sequence has a₁ = 2 and a₄ = −54.
Find the rule for term n.
1 Count the jumps
Term 1 is 2.
Term 4 is −54.
We don't know the terms between.
In a geometric sequence we multiply by the same number r every time.
How many jumps?
4 take away 1: 3 jumps.
Careful!
We count the jumps, not the terms.
4 terms have only 3 jumps between them.
2 Find r
Start at 2, multiply by r 3 times, and we get −54.
r times itself 3 times is r³.
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 r³ is left.
−54 divided by 2 is −27.
Which number times itself 3 times is −27?
−3.
A minus, an odd number of times, stays minus.
So r = −3.
3 Find the first term
Term 1 is given: 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 = −3.
So the rule is aₙ = 2 × (−3)ⁿ⁻¹.
5 Check
Let's check with the terms we were given.
Put n = 1: the power is 0.
We get 2.
And n = 4 gives −54.
Both work!
Let's look back.
First: count the jumps.
Then: find r.
Then: find the first term.
Then: write the rule.
Last: check.