Step by step
The question
Question: find the sum 4 + 8 + 12 + and so on + 80.
1 Find the difference
2 Count the terms
Careful!
(80 − 4)/4 counts the jumps.
There is one more term than jumps.
The rule from the sequences lesson: aₙ = 4n.
The last term, 80, is term number n.
So we solve 4n = 80.
So 4n = 80.
4n means 4 times n.
We want just n.
To remove times 4, we divide by 4.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
4 divided by 4 is 1.
So just n is left.
80 divided by 4 is 20.
So n = 20!
So there are 20 terms.
3 Add in pairs
The pairs rule from the sequences lesson: first plus last makes every pair.
Times the number of terms, over 2.
Put in n = 20, the first term 4 and the last term 80.
4 + 80 is 84.
20 times 84 is 1680.
Half of that is 840.
4 Check
Let's check the pairs: 4 + 80 is 84.
The second and the second-last: 8 + 76 is 84 too.
20 terms make 10 pairs.
10 times 84 is 840.
It works!
Let's look back.
First: find the difference.
Then: count the terms.
Then: add in pairs.
Last: check.