Step by step
The question
Question: 2, 10, 50, …: find the sum of the first 4 terms.
A sequence is a list of numbers in order.
1 Find the ratio
We multiply by 5 every time.
A list like this is a geometric sequence.
The number we multiply by is called the ratio: r = 5.
2 Write S and 5S
S is the sum: all 4 terms added.
Now times 5: each term times 5 is the next term.
So the sum moves one place to the right.
3 Take away
We take S away from 5S.
The terms in the middle are in both rows.
They cross out.
Only 1250 and 2 are left.
5S take away S is 4S.
1250 − 2 is 1248.
To remove times 4, we divide by 4.
So the sum of the first 4 terms is 312.
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 4 terms one by one.
We get 312 again.
So the sum is right!
Let's look back.
First: find the ratio.
Then: write S and 5S.
Then: take away.
Last: check.