Arithmetic series

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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): S = a + (a+d) + (a+2d) + ... + (a+(n-1)d)

S = (a+(n-1)d) + (a+(n-2)d)+ ... + (a+d) + a

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)

This equals

2S = n(2a+(n-1)d), so the sum is

Example Problems

Introductory Problems

• 2006 AMC 10A Problem 9

See also

• Series
• Summation