SS- Arithmetic Series Lesson

Arithmetic Series

An arithmetic series is the sum of the first n terms in an arithmetic sequence.

This video explains how this formula was derived by the famous mathematician, Carl Friedrich Gauss. 

Let's look at some more examples.  Find the sum of the arithmetic series.

2+6+10+....+58

Solution: Identify the first and last terms.

$a_1=2;a_n=58$

Now we need to know how many terms are in this sequence.  Use the explicit formula to find n.

$a_n=a_1+d(n-1)$

$58=2+4(n-1)$

$58=2+4n-4$

$58=-2+4n$

$60=4n$

$n=15$

Use the formula to find the sum:  $S_n=n(\frac{a_1+a_n}{2})$

$S_{15}=15(\frac{2+58}{2})=15(\frac{60}{2})=15(30)=450$

Sigma Notation

When the sequence is very long, we can use sigma notation to express our series. Let's look at this sequence and write it in sigma notation.

Arithmetic Series:	Sigma Notation:	Solution:
4 + 8 + 12 + 16 + ... + 40		$S_{10}=10(\frac{4+40}{2})=10(\frac{44}{2})=10(22)=220$

 

This video below will show you more examples. Be sure to take notes as you watch as each example is different than the other.