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Consider $s_n=\{2,1,\frac{4}{5},\frac{5}{7},\frac{2}{3}, \ldots\}$

What are some techniques and tips that I can use to find the rules for such series? I find is extremely hard to do so and would like some help.

EDIT: I'm not necessarily, looking for the rule of the specific example given, but some guidelines on to find rules on my own.

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    $\begingroup$ Such problems are usually ill-posed, but here we have 2/1,3/3,4/5,5/7,6/9, spot the pattern! $\endgroup$
    – Peter
    May 27, 2018 at 21:30
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    $\begingroup$ @Peter Wow I never would have thought about rewriting the rule like that! $\endgroup$
    – rainier
    May 27, 2018 at 21:41
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    $\begingroup$ Notice the proper way to code curly braces in MathJax, as in my edit to this question. $$ s_n = \{2,1,\ldots \} $$ $\endgroup$ May 27, 2018 at 22:53
  • $\begingroup$ Thanks @MichaelHardy :) I couldn't figure it out earlier lol $\endgroup$
    – rainier
    May 27, 2018 at 22:59
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    $\begingroup$ If you search for "pattern sequence" on this site you will find lots of discussions about why the very general question you are asking has not good answer: math.stackexchange.com/search?q=pattern+sequence $\endgroup$ May 28, 2018 at 17:35

1 Answer 1

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To me, it seems that it can be written into $$\left\{\frac{n+1}{2n-1}: n\geq1\right\}.$$

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  • $\begingroup$ But what's the logic and thought process behind finding that rule? How did you find it? $\endgroup$
    – rainier
    May 27, 2018 at 21:38
  • $\begingroup$ It's kind of instinct rather than logic. $\endgroup$
    – Muduri
    May 27, 2018 at 21:41
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    $\begingroup$ Sorry, it's been confusing for me whether natural number starts from $0$ or $1$... $\endgroup$
    – Muduri
    May 27, 2018 at 21:47
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    $\begingroup$ Well, under the condition that you are in Europe perhaps... In fact, it doesn't really mater though, you may subtly adjust the formula. $\endgroup$
    – Muduri
    May 27, 2018 at 21:53
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    $\begingroup$ @rainier whether you start at $0$ or you start at $1$ is usually entirely personal preference in strictly mathematical contexts. In computer science contexts, many (but not all) programming languages have lists and arrays with entries starting with index $0$ which helps reinforce the habit in mathematical contexts as well. If you ever worry that it is unclear whether you want to start from $0$ or $1$ then just explicitly state it. See this related question. $\endgroup$
    – JMoravitz
    May 27, 2018 at 21:58

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