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| − | {{stub}}
| + | #REDIRECT[[Arithmetic sequence]] |
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| − | An '''arithmetic series''' is a sum of consecutive terms in an [[arithmetic sequence]]. For instance,
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| − | <math> 2 + 6 + 10 + 14 + 18 </math>
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| − | is an arithmetic series whose value is 50.
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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):
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| − | S = a + (a+d) + (a+2d) + ... + (a+(n-1)d)
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| − | S = (a+(n-1)d) + (a+(n-2)d)+ ... + (a+d) + a
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| − | Now, adding vertically and shifted over one, we get
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| − | 2S = (2a+(n-1)d)+(2a+(n-1)d)+(2a+(n-1)d)+...+(2a+(n-1)d)
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| − | This equals
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| − | 2S = n(2a+(n-1)d), so the sum is <math>\displaystyle \frac{n}{2} (2a+(n-1)d</math>
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| − | == Example Problems ==
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| − | === Introductory Problems ===
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| − | * [[2006_AMC_10A_Problems/Problem_9 | 2006 AMC 10A Problem 9]]
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| − | == See also ==
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| − | * [[Series]]
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| − | * [[Summation]]
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