12.4. Arithmetic Series http://www.ck12.org
12.4 Arithmetic Series
Here you will learn to compute finite arithmetic series more efficiently than just adding the terms together one at a
time.
While it is possible to add arithmetic series one term at a time, it is not feasible or efficient when there are a large
number of terms. What is a clever way to add up all the whole numbers between 1 and 100?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/62253http://www.youtube.com/watch?v=Dj1JZIdIwwo James Sousa: Arithmetic Series
Guidance
The key to adding up a finite arithmetic series is to pair up the first term with the last term, the second term with the
second to last term and so on. The sum of each pair will be equal. Consider a generic series:
n
i=∑ 1 ai=a^1 +a^2 +a^3 +···an
When you pair the first and the last terms and note thatan=a 1 +(n− 1 )kthe sum is:
a 1 +an=a 1 +a 1 +(n− 1 )k= 2 a 1 +(n− 1 )k
When you pair up the second and the second to last terms you get the same sum:
a 2 +an− 1 = (a 1 +k)+(a 1 +(n− 2 )k) = 2 a 1 +(n− 1 )k
The next logical question to ask is: how many pairs are there? If there arenterms total then there are exactlyn 2 pairs.
Ifnhappens to be even then every term will have a partner andn 2 will be a whole number. Ifnhappens to be odd
then every term but the middle one will have a partner andn 2 will include a^12 pair that represents the middle term
with no partner. Here is the general formula for arithmetic series:
n
i∑= 1 ai=n^2 (^2 a^1 +(n−^1 )k)wherekis the common difference for the terms in the series.
Example A
Add up the numbers between one and ten (inclusive) in two ways.
Solution:One way to add up lists of numbers is to pair them up for easier mental arithmetic.