Step by step
The question
Question: a geometric sequence has a₁ = 3 and a₅ = 48.
Find the rule for term n.
1 Count the jumps
Term 1 is 3.
Term 5 is 48.
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 1: 4 jumps.
2 Find r
Start at 3, multiply by r 4 times, and we get 48.
r times itself 4 times is r⁴.
To remove times 3, we divide by 3.
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.
3 divided by 3 is 1.
So just r⁴ is left.
48 divided by 3 is 16.
Which number times itself 4 times is 16?
2 works.
But −2 works too: a minus, an even number of times, is plus.
Careful!
Many people write only r = 2.
They lose r = −2, and its sequence.
So r = 2 or r = −2.
Two ratios, so two sequences.
3 Find the first term
Term 1 is given: 3.
It is the same for both.
4 Write the rules
The rule from the sequences lesson: the first term, times the ratio to the power (n − 1).
So aₙ = 3 × 2ⁿ⁻¹ or aₙ = 3 × (−2)ⁿ⁻¹.
5 Check
Let's check each rule with the terms we were given.
Term 1: 3.
And term 5: 48.
Now the second rule.
Term 1: 3.
And term 5: 48.
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.