Step by step
The question
Question: a geometric sequence has a₁ = 3 and a₄ = 24.
Find the rule for term n.
1 Count the jumps
Term 1 is 3.
Term 4 is 24.
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 3, multiply by r 3 times, and we get 24.
r times itself 3 times is r³.
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 r³ is left.
24 divided by 3 is 8.
Which number times itself 3 times is 8?
2.
So r = 2.
3 Find the first term
Term 1 is given: 3.
4 Write the rule
The rule from the sequences lesson: the first term, times the ratio to the power (n − 1).
Put in a₁ = 3 and r = 2.
So the rule is aₙ = 3 × 2ⁿ⁻¹.
5 Check
Let's check with the terms we were given.
Put n = 1: the power is 0.
We get 3.
And n = 4 gives 24.
Both work!
Let's look back.
First: count the jumps.
Then: find r.
Then: find the first term.
Then: write the rule.
Last: check.