Step by step
The question
Question: 1, 5, 9, 13, …: find the sum of the first 20 terms.
A sequence is a list of numbers in order.
1 Find the difference
We add 4 every time.
A list like this is an arithmetic sequence.
The number we add is called the difference: d = 4.
2 Find the last term
The last term is term 20.
To get there, we add 4, 19 times.
Brackets come first.
20 minus 1 is 19.
Times comes before plus.
19 times 4 is 76.
1 plus 76 is 77.
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
3 Add in pairs
S is the sum: all 20 terms added.
Now the same sum backwards: the last term first.
Add each column.
1 + 77 is 78.
5 + 73 is 78.
Every column makes 78!
There are 20 columns.
So the two sums make 20 times 78.
That is 1560.
One sum is half of that.
We divide by 2.
So the sum of the first 20 terms is 780.
This works for every arithmetic sequence: the number of terms, times first plus last, over 2.
4 Check
Let's check with the rule.
We get 780 again.
So the sum is right!
Let's look back.
First: find the difference.
Then: find the last term.
Then: add in pairs.
Last: check.