Step by step
The question
Question: 4, 12, 36, …: find the sum of the first 5 terms.
A sequence is a list of numbers in order.
1 Find the ratio
We multiply by 3 every time.
A list like this is a geometric sequence.
The number we multiply by is called the ratio: r = 3.
2 Write S and 3S
S is the sum: all 5 terms added.
Now times 3: each term times 3 is the next term.
So the sum moves one place to the right.
3 Take away
We take S away from 3S.
The terms in the middle are in both rows.
They cross out.
Only 972 and 4 are left.
3S take away S is 2S.
972 − 4 is 968.
To remove times 2, we divide by 2.
So the sum of the first 5 terms is 484.
This works for every geometric sequence: the first term, times r to the power n, minus 1, over r minus 1.
4 Check
Let's check: add the 5 terms one by one.
We get 484 again.
So the sum is right!
Let's look back.
First: find the ratio.
Then: write S and 3S.
Then: take away.
Last: check.