A sequence is a list of numbers. A series is created by adding terms in the sequence. This lesson assumes that you know about arithmetic sequences, how to find the common difference and how to find an explicit formula. You may want to review the basics of arithmetic sequences or finding formulas. There are two ways to indicate that you are adding terms in a sequence. One is by using summation notation and one is by using subscript notation similar to how we write explicit forms of sequences. Let’s look at each of these methods separately. First, summation notation. In summation notation, you are given an expression and told how many terms you are to add up.
tells you to use the values of n = 1, n = 2, and n = 3 in the expression 2n + 1.
Once those values are substituted in, you add them all up.
Let's Practice summation notation: Evaluate This problem is asking us to add up five terms. So use n = 1, 2, 3, 4, and 5 in the expression 3n +5.  The series created by adding the first five terms of the sequence in the previous example can be found using another method called subscript notation.
Let's Practice subscript notation:
i.  Find for the sequence .
The formula says we need to know n, the first term, and the n^{th} term. n = 10 since we are asked to find the sum of the first 10 terms
Substituting these values into the equation gives  ii.  Find the sum of the first 21 terms of the sequence 3, 7, 11, 15, . . .
The formula says we need to know n, the first term, and the n^{th} term. n = 21 since we are asked to find the sum of the first 21 terms.
To find a_{1} and a_{21} we will need the explicit formula. So now we can find
We now have everything we need to find the sum of the first 21 terms. 


