Arithmetic series

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An arithmetic series is a sum of consecutive terms in an arithmetic sequence. For instance,

$2 + 6 + 10 + 14 + 18$

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):

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

Now, adding vertically and shifted over one, we get

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

This equals $\displaystyle 2S = n(2a+(n-1)d)$ , so the sum is $\displaystyle \frac{n}{2} (2a+(n-1)d$ .

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Arithmetic series

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