Step by step
The question
Question: find the sum 81 + 27 + 9 + and so on + 1.
1 Find the ratio
2 Count the terms
The rule from the sequences lesson.
The last term is term n.
So 81 times (1/3) to the power (n − 1) is 1.
To remove times 81, we divide by 81.
1 divided by 81 is 1/81.
Which power of (1/3) is 1/81?
Multiply (1/3) by itself 4 times: we get 1/81.
So the power is 4.
So n − 1 = 4.
The power counts the jumps.
Add 1 for the first term: n = 5 terms.
Careful!
The power 4 counts the jumps.
There is one more term than jumps: 5 terms.
3 Use the sum rule
r is less than 1, so we use the same rule written the other way round: 1 minus r to the power n, over 1 minus r.
Nothing negative.
Put in a₁ = 81, r = 1/3 and n = 5.
(1/3) to the power 5 is 1/243.
1 take away that is 242/243.
And 1 − (1/3) is 2/3.
So the top is the first term times that.
Then we divide by the bottom.
Dividing by 2/3 is times its flip, 3/2.
So the sum is 121.
4 Check
Let's check: we add the 5 terms one by one.
We get 121 again.
It works!
Let's look back.
First: find the ratio.
Then: count the terms.
Then: use the sum rule.
Last: check.