Difference between revisions of "Arithmetic series"

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 in two as so ($S$ is the sum, $a$ is the first term, $z$ is the number of terms, and $d$ is the common difference): $$S = a + (a+d) + (a+2d) + \ldots + (z-d) + z$$ Flipping the right side of the equation we get $$S = z + (z-d) + (z-2d) + \ldots + (a+d) + a$$

Now, adding the above two equations vertically, we get

$$2S = (a+z) + (a+z) + (a+z) + ... +(a+z) + (a+z)$$

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