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Arithmetic Series
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.
  1. 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.
Summary
In general, summation notation looks like where
an is an expression with n as the variable and
k tells you how many terms to add.


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.
  1. 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.
Summary

To find the sum of terms in an arithmetic sequence, use the following formula.
In this formula:
Sn is the sum of the first n terms in a sequence
n is the number of terms you are adding up
a1 is the first term of the sequence
an is the nth term of the sequence
The nth term is found by using the explicit formula for the sequence. So if you are not given the explicit formula, you will need to find it before you can find the sum.



Let's Practice subscript notation:

i.  Find for the sequence .

The formula says we need to know n, the first term, and the nth 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 nth term.
n = 21 since we are asked to find the sum of the first 21 terms.

To find a1 and a21 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.

Examples
Example Evaluate
What is your answer?
 
Example Evaluate
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Example Find for
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Example Find for
What is your answer?
 
Example Find the sum of the first 100 terms of 5, 11, 17, 23, . . .
What is your answer?
 
Example Find the sum of the first 73 terms of 12, 20, 28, 36, . . .
What is your answer?
 



S Taylor

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