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 as so (S is the sum, a is the first term, n is the number of terms, and d is the common difference):
 
<cmath>\begin{align*}
 
S &=  a + (a+d) + (a+2d) + ... + (a+(n-1)d) \
 
S &= (a+(n-1)d) + (a+(n-2)d)+ ... + (a+d) + a
 
\end{align*}</cmath>
 
 
 
Now, adding vertically and shifted over one, we get
 
 
 
<cmath>2S = (2a+(n-1)d)+(2a+(n-1)d)+(2a+(n-1)d)+...+(2a+(n-1)d)</cmath>
 
 
 
This equals <math>2S = n(2a+(n-1)d)</math>, so the sum is <math>\frac{n}{2} (2a+(n-1)d)</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