# Methods for finding rules for ${s_n}$

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.

• Such problems are usually ill-posed, but here we have 2/1,3/3,4/5,5/7,6/9, spot the pattern! May 27, 2018 at 21:30
• @Peter Wow I never would have thought about rewriting the rule like that! May 27, 2018 at 21:41
• Notice the proper way to code curly braces in MathJax, as in my edit to this question. $$s_n = \{2,1,\ldots \}$$ May 27, 2018 at 22:53
• Thanks @MichaelHardy :) I couldn't figure it out earlier lol May 27, 2018 at 22:59
• 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 May 28, 2018 at 17:35

## 1 Answer

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

• But what's the logic and thought process behind finding that rule? How did you find it? May 27, 2018 at 21:38
• It's kind of instinct rather than logic. May 27, 2018 at 21:41
• Sorry, it's been confusing for me whether natural number starts from $0$ or $1$... May 27, 2018 at 21:47
• Well, under the condition that you are in Europe perhaps... In fact, it doesn't really mater though, you may subtly adjust the formula. May 27, 2018 at 21:53
• @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. May 27, 2018 at 21:58