Difference between revisions of "Arithmetic series"

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An '''arithmetic series''' is a sum of consecutive terms in an [[arithmetic sequence]].  For instance,
+
#REDIRECT[[Arithmetic sequence]]
 
 
<math> 2 + 6 + 10 + 14 + 18 </math>
 
 
 
is an arithmetic series whose value is 50.
 
 
 
To find the sum of an arithmetic sequence, we can write it out in two as so (<math>S</math> is the sum, <math>a</math> is the first term, <math>z</math> is the number of terms, and <math>d</math> is the common difference):
 
<cmath>
 
S =  a  + (a+d) + (a+2d) + \ldots  + (z-d)  + z
 
</cmath>
 
Flipping the right side of the equation we get
 
<cmath>
 
S = z  + (z-d) + (z-2d) + \ldots  +  (a+d)    + a
 
</cmath>
 
 
 
Now, adding the above two equations vertically, we get
 
 
 
<cmath>2S = (a+z) + (a+z) + (a+z) + ... +(a+z) + (a+z)</cmath>
 
 
 
This equals <math>2S = n(a+z)</math>, so the sum is <math>\frac{n(a+z)}{2}</math>.
 
 
 
== Problems ==
 
=== Introductory Problems ===
 
* [[2006_AMC_10A_Problems/Problem_9 | 2006 AMC 10A, Problem 9]]
 
*[[2006 AMC 12A Problems/Problem 12 | 2006 AMC 12A, Problem 12]]
 
 
 
=== Intermediate Problems ===
 
*[[2003 AIME I Problems/Problem 2|2003 AIME I, Problem 2]]
 
 
 
=== Olympiad Problem ===
 
 
 
== See also ==
 
* [[Series]]
 
* [[Summation]]
 
 
 
{{stub}}
 

Latest revision as of 10:29, 31 August 2021