Arithmetic progressions (or arithmetic sequences) are sequences with a common difference (i.e. the same number is added or subtracted to get the next number in the sequence).
For example,
or
The 
term of an arithmetic progression is 
where 
is the first term and 
is the common difference.
i.e. For the sequence above, 
An arithmetic series is the sum of the arithmetic progression.
For example, if the sequence is
then 
The series is also a sequence and we are going to find the general term, 
.
which we can write as
Now, I am going to write that in reverse order (to make the next bit more obvious)
I am going to add the two versions of together
Each term has an and there are 
terms, so we now have 
. The terms, we going to group together
Which simplifies to 
and we have 
terms. So this part of the sum is 
Thus we have
Which simplifies to
(1) 
Let’s test it, remember the sequence . We know 
and 
, then prove that 
. Here where 
and 
are terms of an arithmetic sequence.
then,
from equation 
into equation 
