Step by step
The question
Question: a geometric sequence has a₂ = 6 and a₄ = 54.
Find the rule for term n.
1 Count the jumps
Term 2 is 6.
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 2: 2 jumps.
2 Find r
Start at 6, multiply by r 2 times, and we get 54.
r times itself 2 times is r².
To remove times 6, we divide by 6.
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.
6 divided by 6 is 1.
So just r² is left.
54 divided by 6 is 9.
Which number times itself 2 times is 9?
3 works.
But −3 works too: a minus, an even number of times, is plus.
Careful!
Many people write only r = 3.
They lose r = −3, and its sequence.
So r = 3 or r = −3.
Two ratios, so two sequences.
3 Find the first term
a₁ is 1 jump before a₂.
Going back, we divide by r.
With r = 3: 2.
With r = −3: −2.
4 Write the rules
The rule from the sequences lesson: the first term, times the ratio to the power (n − 1).
So aₙ = 2 × 3ⁿ⁻¹ or aₙ = −2 × (−3)ⁿ⁻¹.
5 Check
Let's check each rule with the terms we were given.
Term 2: 6.
And term 4: 54.
Now the second rule.
Term 2: 6.
And term 4: 54.
Both rules work!
Let's look back.
First: count the jumps.
Then: find r.
Then: find the first term.
Then: write the rules.
Last: check.