**Date:** January 4 2021

**Summary:** A brief overview on what arithmetic series are and some of its underlying math.

**Keywords:** ##zettel #arithmetic #sums #series #proof #archive

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An arithmetic sequence is one which the difference between one term and the next only differs by a constant. One example is this:

$1+2+3+\cdots+n$where the difference between each proceeding term is the constant value, 1.

An arithmetic series is one in which values in an arithmetic sequence are summed together:

$\sum_{k=1}^{n} k=1+2+\cdots+n$This is a visual proof of the Arithmetic Series algorithm:

A formalization of the above is:

$\sum_{k=1}^{n} = \frac{n(a_{1} + a_{n})}{2}$which is equivalent to:

$\sum_{k=1}^{n} = \frac{a_{n}(a_{1} + a_{n})}{2}$The latter formalization is somewhat more common and it works as $a_{n}$ gives the same values as what the size of the sequence is which is $n$. From the visual proof, the $\frac{n}{2}$ constant comes from halving the size of each region.

(Thanks to Mark Kittisopikul, Yingbo Ma, and Benoit Pasquier for these explanations)

Zelko, Jacob. *Arithmetic Sums & Series*. https://jacobzelko.com/01052021044121-arithmetic-series. January 4 2021.

CC BY-SA 4.0 Jacob Zelko. Last modified: July 16, 2023.
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