I recently came across a very old Algebra textbook from the 1860s, and on the chapter discussing "arithmetical progression", it says there are "20 cases for arithmetical progression". The book then shows a table with 20 different formulae for arithmetic progressions. My question is this: I have looked in many other new textbooks and have only seen 1 or 2 basic formulae for arithmetic progression but no where near as many as in the 1860s textbook. Am I missing something or were 19th century textbooks in Math much more rigorous and detailed? I'll give the first 4 formulae listed:

  1. Given: a, d, n l = a + (n - 1)d
  2. Given: a, d, S l = -1/2d±$\sqrt{2dS + (a - 1/2d)^2}$
  3. Given: a, n, S l = $\frac{2S}{N}$ - a
  4. Given: d, n, S l = $\frac{S}{n}$ + $\frac{(n-1)d}{2}$

I have never seen it set up this way. Any thoughts? They go on to list 15 more forms!

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    $\begingroup$ I sure wouldn't want students thinking they had to memorize all those formulas. Do any of them seem particularly useful to you? I would not call this more rigorous. $\endgroup$
    – Sue VanHattum
    Jun 3 '21 at 21:09
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    $\begingroup$ Something seems to be missing in these formulas, such as what is being summed. i.e. a linear progression, squares, etc. These types of things tend to be taught where they are useful, nowadays, not as isolated facts. $\endgroup$
    – Adam
    Jun 3 '21 at 22:36
  • $\begingroup$ Also, why just 20? $\endgroup$
    – Adam
    Jun 3 '21 at 22:38
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    $\begingroup$ Adam: The book makes no mention of what is being summed. It just puts this exact thing in a huge table with the 20 cases. The book affirms there are "20 cases" but does not say why. $\endgroup$
    – Wasp
    Jun 3 '21 at 23:07
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    $\begingroup$ I agree with @Nij. those 20 formulas seem to contain just two equations' worth of information, and one of those two is the definition of "arithmetic progression". Memorizing all 20 equations seems a waste of time and likely to lead to errors (because most people's memory isn't good enough to retain that much unmotivated and incoherent-looking information). The only possible value I see here would be training the students' memories, and even for that I'd rather have them memorize some poetry. $\endgroup$ Jun 4 '21 at 17:14

These are the first term, last term, difference between successive terms, number of terms being counted from first, and sum from first to last term of an arithmetic sequence.

The formulae allow for finding each given the others (there are five things to find and four ways to pick three of the rest as required). They are all rearrangements of the basic formulae for the progression and the sum.

At least in theory, modern algebra classes try to teach the understanding of what creates the rules, not just memorise the rules themselves, so that students today need only two formulae and can reinvent whatever form they need.

  • $\begingroup$ Thank you for the input. Here are some more formulae they list: S = (1 + a)/2 * n and 1/2n[2a + (n - 1)d]. $\endgroup$
    – Wasp
    Jun 4 '21 at 21:47
  • $\begingroup$ That first formula should be an l, not a 1, for "last term". $\endgroup$
    – Nij
    Jun 4 '21 at 21:54

I'm not that crazy about the long list on that one topic. The one potential benefit would be to have kids derive each from the definition of arithmetic progression. Sort of just getting practice in manipulation.

Maybe a little similar to how playing with different trig identities is useful. Nobody needs to memorize all of them. But knowing a few and deriving others is a good muscle-building activity. And I'm not saying it has to be the bare minimum. But probably much more towards the minimum than the converse and emphasis on deriving others as needed. For sure, the practice in manipulation is helpful later in puzzle solving activities of anti-differentiation.


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