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, | An '''arithmetic series''' is a sum of consecutive terms in an [[arithmetic sequence]]. For instance, | ||
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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): | 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): | ||
| − | S = a + (a+d) + (a+2d) + ... + (a+(n-1)d) | + | <div style="text-align:center;"><math>\displaystyle S = a + (a+d) + (a+2d) + ... + (a+(n-1)d)</math> |
| − | S = (a+(n-1)d) + (a+(n-2)d)+ ... + (a+d) + a | + | <math>\displaystyle S = (a+(n-1)d) + (a+(n-2)d)+ ... + (a+d) + a</math></div> |
Now, adding vertically and shifted over one, we get | Now, adding vertically and shifted over one, we get | ||
| − | 2S = (2a+(n-1)d)+(2a+(n-1)d)+(2a+(n-1)d)+...+(2a+(n-1)d) | + | <div style="text-align:center;"><math>\displaystyle 2S = (2a+(n-1)d)+(2a+(n-1)d)+(2a+(n-1)d)+...+(2a+(n-1)d)</math></div> |
| − | This equals | + | This equals <math>\displaystyle 2S = n(2a+(n-1)d)</math>, so the sum is <math>\displaystyle \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 == | == See also == | ||
* [[Series]] | * [[Series]] | ||
* [[Summation]] | * [[Summation]] | ||
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| + | {{stub}} | ||
Revision as of 14:55, 12 March 2007
An arithmetic series is a sum of consecutive terms in an arithmetic sequence. For instance,
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):
Now, adding vertically and shifted over one, we get
This equals 
, so the sum is 
.
Contents
Problems
Introductory Problems
Intermediate Problems
Olympiad Problem
See also
This article is a stub. Help us out by expanding it.
