Step by step
The question
Question: find the sum 27 + 18 + 12 + 8 + and so on.
The dots at the end mean: it goes on forever.
1 Find the ratio
2 Look at the sums
2/3 is less than 1.
Each new term is smaller.
So the sums settle down near one number.
Let's add the terms one at a time, and write each running sum: 27, 45, 57, and so on.
3 Use the sum rule
In the sum rule, r to the power n gets closer and closer to 0.
So it drops out, and the sum is the first term over 1 minus r.
Careful!
The terms go on forever, but the sum does not: each term is smaller than the one before.
Put in a₁ = 27 and r = 2/3.
1 − 2/3 is 1/3.
So we divide by 1/3.
Dividing by 1/3 is times its flip, 3.
So the sum is 81.
4 Check
Let's check: we keep adding terms.
The running sums are 73.889, 76.259, 77.84.
They get closer and closer to 81.
It works!
Let's look back.
First: find the ratio.
Then: look at the sums.
Then: use the sum rule.
Last: check.