Step by step
The question
Question: find the sum 96 + 48 + 24 + and so on + 3.
1 Find the ratio
2 Count the terms
The rule from the sequences lesson.
The last term is term n.
So 96 times (1/2) to the power (n − 1) is 3.
To remove times 96, we divide by 96.
3 divided by 96 is 1/32.
Which power of (1/2) is 1/32?
Multiply (1/2) by itself 5 times: we get 1/32.
So the power is 5.
So n − 1 = 5.
The power counts the jumps.
Add 1 for the first term: n = 6 terms.
Careful!
The power 5 counts the jumps.
There is one more term than jumps: 6 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 = 96, r = 1/2 and n = 6.
(1/2) to the power 6 is 1/64.
1 take away that is 63/64.
And 1 − (1/2) is 1/2.
So the top is the first term times that.
Then we divide by the bottom.
Dividing by 1/2 is times its flip, 2.
So the sum is 189.
4 Check
Let's check: we add the 6 terms one by one.
We get 189 again.
It works!
Let's look back.
First: find the ratio.
Then: count the terms.
Then: use the sum rule.
Last: check.