Step by step
The question
Question: a geometric sequence has a₃ = 12 and a₄ = 24.
Find the rule for term n.
1 Count the jumps
Term 3 is 12.
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 3: 1 jump.
2 Find r
Start at 12, multiply by r 1 times, and we get 24.
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 r¹ is left.
24 divided by 12 is 2.
So r = 2.
3 Find the first term
a₁ is 2 jumps before a₃.
Going back, we divide by r each time.
2² is 4.
12 divided by 4 is 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 = 3: the power is 2.
We get 12.
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.