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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.

Michael Hardy
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rainier
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    Such problems are usually ill-posed, but here we have 2/1,3/3,4/5,5/7,6/9, spot the pattern! – Peter May 27 '18 at 21:30
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    @Peter Wow I never would have thought about rewriting the rule like that! – rainier May 27 '18 at 21:41
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    Notice the proper way to code curly braces in MathJax, as in my edit to this question. $$ s_n = {2,1,\ldots } $$ – Michael Hardy May 27 '18 at 22:53
  • Thanks @MichaelHardy :) I couldn't figure it out earlier lol – rainier May 27 '18 at 22:59
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    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: https://math.stackexchange.com/search?q=pattern+sequence – Ethan Bolker May 28 '18 at 17:35

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

Michael Hardy
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Muduri
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  • But what's the logic and thought process behind finding that rule? How did you find it? – rainier May 27 '18 at 21:38
  • It's kind of instinct rather than logic. – Muduri May 27 '18 at 21:41
  • Dang, so it's not something I can learn how to do? :( – rainier May 27 '18 at 21:43
  • Also wait... $\frac{1+2}{2(1)-1}=3$...Am I that sleep deprived or is it not the correct rule. When n=1, it should be 2... – rainier May 27 '18 at 21:44
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    Sorry, it's been confusing for me whether natural number starts from $0$ or $1$... – Muduri May 27 '18 at 21:47
  • Under what conditions am I allowed to start at 0? Sequences and series in my textbook almost always start at n=1. Therefore, starting at 0 wouldn't be valid for many problems. – rainier May 27 '18 at 21:50
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    Well, under the condition that you are in Europe perhaps... In fact, it doesn't really mater though, you may subtly adjust the formula. – Muduri May 27 '18 at 21:53
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    @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. – JMoravitz May 27 '18 at 21:58
  • @JMoravitz Noted! Thanks, this isn't the first time I got hung up over trivial things. – rainier May 27 '18 at 22:00