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Arithmetic Sums & Series

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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Table of Contents

  1. How To Cite
  2. References
  3. Discussion:

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++n 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:

k=1nk=1+2++n \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:

k=1n=n(a1+an)2 \sum_{k=1}^{n} = \frac{n(a_{1} + a_{n})}{2}

which is equivalent to:

k=1n=an(a1+an)2 \sum_{k=1}^{n} = \frac{a_{n}(a_{1} + a_{n})}{2}

The latter formalization is somewhat more common and it works as ana_{n} gives the same values as what the size of the sequence is which is nn. From the visual proof, the n2\frac{n}{2} constant comes from halving the size of each region.

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

How To Cite

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

References

Discussion:

CC BY-SA 4.0 Jacob Zelko. Last modified: July 16, 2023. Website built with Franklin.jl and the Julia programming language.