Step by step
The question
Question: 7, 3, −1, −5, …: find the sum of the first 8 terms.
A sequence is a list of numbers in order.
1 Find the difference
We take away 4 every time.
A list like this is an arithmetic sequence.
The number we add is called the difference: d = −4.
It is negative, because we go down.
2 Find the last term
The last term is term 8.
To get there, we add (−4), 7 times.
Brackets come first.
8 minus 1 is 7.
Times comes before plus.
Plus times minus is minus.
7 times 4 is 28.
So we get −28.
Adding a negative is the same as taking away.
So 7 + (−28) becomes 7 − 28.
We take away more than we have.
Start at 7.
Go down 28.
We pass 0 and land on −21.
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
3 Add in pairs
S is the sum: all 8 terms added.
Now the same sum backwards: the last term first.
Add each column.
7 + (−21) is −14.
3 + (−17) is −14.
Every column makes −14!
There are 8 columns.
So the two sums make 8 times −14.
That is −112.
One sum is half of that.
We divide by 2.
So the sum of the first 8 terms is −56.
This works for every arithmetic sequence: the number of terms, times first plus last, over 2.
4 Check
Let's check: add the 8 terms one by one.
We get −56 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.